一类复合泊松过程的首达时概率密度

一类复合泊松过程的首达时概率密度 On the First Passage Time Density for a Class of Compound Poisson Process first-author 陶秀丽，女，硕士生，现从事随机过程研究。E-mail：taoxiuli@whu.edu.cn 陶秀丽，女，硕士生，现从事随机过程研究。E-mail：taoxiuli@whu.edu.cn taoxiuli@whu.edu.cn corresp E-mail： shenzh@whu.edu.cn E-mail： shenzh@whu.edu.cn shenzh@whu.edu.cn 武汉大学 数学与统计学院，湖北 武汉430072 武汉大学 数学与统计学院，湖北 武汉430072 School of Mathematics and Statistics， Wuhan University， Wuhan 430072， Hubei， China School of Mathematics and Statistics， Wuhan University， Wuhan 430072， Hubei， China 利用复合泊松过程的独立增量性，研究了一类带正跳跃的复合泊松过程首达时的概率密度函数，这类复合泊松过程的跳跃幅度服从有限离散分布。由首达时的概率密度函数，计算出首达时有限的概率。而后，作为主要结论的应用，计算了几个特殊例子，推广了关于独立泊松过程的加权和过程的首达时概率分布的一些结论。 Using the independent increment property of compound Poisson process, we studied the probability density function of the first passage time of a class of compound Poisson processes with positive jumps. The jump size of those compound Poisson processes satisfies finite discrete distribution. From the probability density function of the first passage time, we derived the probability of which first passage time is finite. Then, as the applications of the main results of this article, several special cases were discussed, and some results about the probability distribution of the first passage time of weighted Poisson processes were generalized. 复合泊松过程 首达时 概率密度函数 compound Poisson process first passage time probability density function 0　 引 言 1 复合泊松过程是一类常见的随机过程，具有平稳增量性，其作为计数过程中一种重要的模型，常用于描述单位时间（空间）内随机事件发生的次数。在日常生活中，常常需要考虑这样一些问题，在固定的一段时间内，到达商店的顾客数或某电话总机接到的呼唤次数；在电子技术领域中的散粒噪声和脉冲噪声；数字通信中已编码信号的误码数等。在这些问题的研究中，使用复合泊松过程建模是一种常用方法。 首达时是一种非常重要的停时，在随机过程理论和应用中都有十分重要的意义。关于连续时间马尔可夫过程首达时概率的相关计算，历史上有过很多结论。文献［ 1 1 ， 2 2 ］研究了布朗运动在与时间相关的某点处的首达时的概率密度，文献［ 3 3 ， 4 4 ］进一步验证了布朗运动在固定点的首达时的概率密度可由布朗运动的强马尔可夫性质得到，文献［ 5 5 ~ 7 7 5~7 ］则研究了O-U过程的首达时的概率分布。更进一步地，文献［ 8 8 ］研究了满足Cherkasov条件的Ito扩散过程的首达时的概率分布。在带跳过程的首达时分布方面的研究相对较少，文献［ 9 9 ］给出了一类带非负跳的Lévy过程在固定点首达时的概率密度函数。文献［ 10 10 ］研究了 n 阶的从属泊松过程的分布性质、长期依赖关系和极限定理，文献［ 11 11 ］研究了一些独立同参数泊松过程加权和的首达时概率密度函数，而后文献［ 12 12 ］把［ 11 11 ］的结论推广到了各泊松过程参数不同的情形。而文献［ 11 11 ， 12 12 ］中的情形均为复合泊松过程的特殊情况，因此这些结论可以进一步推广。 本文将沿用文献［ 11 11 ， 12 12 ］中的方法，研究如下带正跳跃的复合泊松过程 <math display="block" id="M1"><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></munderover></mstyle><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524785&type= 21.67466736 8.80533314 的首达时的密度函数 <math id="M2"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524799&type= 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2.79399991 点的首达时为 <math id="M6"><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>≜</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mo stretchy="false">{</mo><mi>s</mi><mo>:</mo><mi>s</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mo stretchy="false">}</mo><mo>,</mo><mi>b</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524827&type= 55.71066666 4.74133348 。当 <math id="M7"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524834&type= 1.86266661 2.79399991 不属于 <math id="M8"><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524894&type= 2.70933342 3.21733332 … <math id="M15"><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524902&type= 2.70933342 3.21733332 <math id="M16"><msub><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524910&type= 2.70933342 3.89466667 <math id="M17"><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524922&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524918&type= 2.70933342 3.89466667 <math 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524950&type= 34.29000092 3.72533321 ， <math id="M21"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524958&type= 44.02666855 4.74133348 ，且 <math id="M22"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524966&type= 14.64733315 8.38199997 。 <math id="M23"><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524974&type= 8.04333305 3.55599999 是参数为 <math id="M24"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524982&type= 1.77800000 2.79399991 的泊松过程，且与 <math id="M25"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><mi>m</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524989&type= 17.35666656 4.91066647 序列之间是相互独立的。而后，本文将计算几个特例，并且对文献［ 11 11 ， 12 12 ］中的相应结论进行补充和推广。 1　 <math id="M26"><mi>X</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524995&type= 8.29733276 4.57200003 的首达时概率密度的计算 1 为了计算 <math id="M27"><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525000&type= 7.70466709 3.55599999 的首达时概率密度，需要利用以下两个引理。 引理 1 定义无穷小时间内增量 <math id="M28"><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>≜</mo><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo>+</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525005&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525004&type= 41.82533264 3.81000018 ，则 <math id="M29"><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525007&type= 9.56733322 3.55599999 的分布为 <math display="block" id="M30"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525011&type= 66.20932770 10.15999985 证 由 <math id="M31"><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525016&type= 9.56733322 3.55599999 的定义可知 <math display="block" id="M32"><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mn 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525021&type= 48.09066772 10.58333302 根据泊松过程的性质， <math id="M33"><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>=</mo><mi>λ</mi><mi mathvariant="normal">d</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525025&type= 31.91933250 3.55599999 。由于对任意 <math id="M34"><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525030&type= 7.95866680 2.79399991 ， <math id="M35"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>N</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525034&type= 9.56733322 3.89466667 与 <math id="M36"><mi>N</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525040&type= 9.82133293 3.55599999 独立，因此有 <math display="block" id="M37"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><mi>N</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>N</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><mi>N</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>N</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">d</mi><mi>t</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo/></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525048&type= 103.12400055 17.35666656 以及 <math display="block" id="M38"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mfenced open="|" close="" separators="|"><mrow><mi>N</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>λ</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>N</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mi>λ</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>λ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo/></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525052&type= 123.61333466 11.09133339 故 <math id="M39"><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525056&type= 9.56733322 3.55599999 的分布即为（2）式。证毕。 引理2 给定 <math id="M40"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525062&type= 1.10066664 2.79399991 ，随机变量 <math id="M41"><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525067&type= 7.70466709 3.55599999 的分布为 <math display="block" id="M42"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msup></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>t</mi></mrow></msup><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>k</mi></mtd></mtr></mtable></mrow></munder><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow><mo>,</mo><mi>k</mi><mo>></mo><mn mathvariant="normal">0</mn></mrow></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525071&type= 89.66200256 19.98133469 这里约定，如果满足 <math id="M43"><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525076&type= 82.80400085 4.48733330 的数组 <math id="M44"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525079&type= 23.70666695 5.67266655 不存在，则 <math id="M45"><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>k</mi></mtd></mtr></mtable></mrow></munder><mrow><mo>⋯</mo></mrow><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525084&type= 35.81399918 11.17599964 ，下文同。 证 利用全概率公式，可得 <math display="block" id="M46"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></munderover></mstyle><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">|</mo><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>t</mi></mrow></msup><msubsup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>t</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>t</mi><mo>-</mo><mi>λ</mi><mi>t</mi></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo/></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525088&type= 111.25200653 30.14133263 当 <math id="M47"><mi>k</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525092&type= 8.38199997 2.79399991 时，同样利用全概率公式，有 <math display="block" id="M48"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>P</mi><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></munderover></mstyle><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mo>=</mo><mi>k</mi><mo stretchy="false">|</mo><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></munderover></mstyle><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mo>=</mo><mi>k</mi><mo stretchy="false">|</mo><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>t</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>t</mi></mrow></msup><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi></mtd></mtr></mtable></mrow></munder><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>t</mi></mrow></msup><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi></mtd></mtr></mtable></mrow></munder><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>t</mi></mrow></msup><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>k</mi></mtd></mtr></mtable></mrow></munder><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow><mtext> </mtext><mi mathvariant="normal">　</mi><mi mathvariant="normal">　</mi><mi mathvariant="normal">　</mi><mi mathvariant="normal">　</mi><mi mathvariant="normal">　</mi><mi mathvariant="normal">　</mi><mi mathvariant="normal">　</mi><mi mathvariant="normal">　</mi><mi mathvariant="normal">　</mi><mi mathvariant="normal">　</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525095&type= 138.26066589 51.13866806 证毕。 利用引理1和引理2，可以得出本文的主要结论，即下面的定理。 定理1 <math id="M49"><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525097&type= 7.70466709 3.55599999 在 <math id="M50"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525101&type= 1.86266661 2.79399991 点的首达时 <math id="M51"><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525108&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525106&type= 3.72533321 3.72533321 的概率密度函数为 <math display="block" id="M52"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi><mo>,</mo><mi mathvariant="normal">若</mi><mo>∃</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn><mi mathvariant="normal">使</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">b</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>,</mo><mi mathvariant="normal">若</mi><mi mathvariant="normal">b</mi><mo>></mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525110&type= 152.65400696 38.86199951 证 当存在 <math id="M53"><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525116&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525114&type= 3.04799986 3.72533321 使得 <math id="M54"><mi>b</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525118&type= 10.41399956 4.06400013 时，根据复合泊松过程的独立增量性，以及 <math id="M55"><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525124&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525122&type= 7.70466709 3.55599999 的非降性，有 <math display="block" id="M56"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>,</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">)</mo><mtext> </mtext><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525128&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525125&type= 147.91265869 9.22866726 由引理1可知 <math id="M57"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub><mi mathvariant="normal">d</mi><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525130&type= 35.81399918 4.65666676 ，再根据（3）式可得 <math display="block" id="M58"><mtable columnalign="left"><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></munderover></mstyle><mrow><mi>λ</mi><mi>p</mi></mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo>=</mo></mtd></mtr><mtr><mtd><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo>=</mo></mtd></mtr><mtr><mtd><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525133&type= 111.92933655 40.30133438 当 <math id="M59"><mi>b</mi><mo>></mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525137&type= 9.65199947 3.72533321 时，与上面的证明过程类似，可得 <math display="block" id="M60"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>,</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525141&type= 84.07399750 18.62666702 <math display="block" id="M61"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><mi>s</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525145&type= 113.87666321 18.11866570 证毕。 定理2 <math id="M62"><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525150&type= 7.70466709 3.55599999 在 <math id="M63"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525154&type= 1.86266661 2.79399991 点的首达时 <math id="M64"><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525158&type= 3.72533321 3.72533321 有限的概率为 <math display="block" id="M65"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mrow></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow><mo>,</mo><mi mathvariant="normal">若</mi><mo>∃</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn><mi mathvariant="normal">使</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">b</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mrow></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow><mo>,</mo><mi mathvariant="normal">若</mi><mi mathvariant="normal">b</mi><mo>></mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn><mo>,</mo><mi mathvariant="normal">若</mi><mi mathvariant="normal">b</mi><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">其</mi><mi mathvariant="normal">他</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525161&type= 121.92000580 40.21666718 这里约定 <math id="M66"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></munderover></mstyle><mrow><mo>⋯</mo><mo>=</mo><mn mathvariant="normal">0</mn></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525164&type= 15.15533447 8.38199997 ，下文同。 证 显然 <math id="M67"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525166&type= 24.80733299 4.48733330 。当存在 <math id="M68"><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525168&type= 3.04799986 3.72533321 使得 <math id="M69"><mi>b</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525170&type= 10.41399956 4.06400013 时， <math display="block" id="M70"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow></mrow></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mrow></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525172&type= 114.97733307 42.58733368 当 <math id="M71"><mi>b</mi><mo>></mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525175&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525174&type= 9.65199947 3.72533321 时， <math display="block" id="M72"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>λ</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mrow></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525176&type= 146.21932983 26.92399788 除此以外的其他情况， <math id="M73"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525178&type= 1.86266661 2.79399991 不属于 <math id="M74"><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525180&type= 7.70466709 3.55599999 的状态空间，即 <math id="M75"><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525183&type= 12.36133385 3.72533321 ，故 <math id="M76"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525185&type= 24.80733299 4.48733330 。证毕。 特别地，当 <math id="M77"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525187&type= 3.30199981 4.57200003 的取值中不含0，即 <math id="M78"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525189&type= 9.90599918 4.57200003 ，可得 <math display="block" id="M79"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mrow></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow><mo>,</mo><mi mathvariant="normal">若</mi><mo>∃</mo><mtext> </mtext><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">使</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">b</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mrow></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow><mo>,</mo><mi mathvariant="normal">若</mi><mi mathvariant="normal">b</mi><mo>></mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">其</mi><mi mathvariant="normal">他</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525192&type= 111.16732788 34.79800034 2　 一些特例的计算 1 在本节中，作为定理1和定理2的应用，我们将给出一些特殊例子的 <math id="M80"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525195&type= 16.76399994 4.48733330 或者 <math id="M81"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525198&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525197&type= 18.28800011 4.48733330 的表达式。 例1 （泊松过程） <math id="M82"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525199&type= 3.30199981 4.57200003 服从单点分布， <math id="M83"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525202&type= 22.69066620 4.65666676 ， <math id="M84"><mi>b</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℤ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525205&type= 10.07533360 3.38666677 ，计算 <math id="M85"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525207&type= 16.76399994 4.48733330 。 由定理1可知： <math id="M86"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>λ</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525209&type= 31.75000000 4.65666676 ,以及当 <math id="M87"><mi>b</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525213&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525212&type= 8.46666718 3.30199981 时， <math display="block" id="M88"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>λ</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac></mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>=</mo><mi>λ</mi><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mn mathvariant="normal">1</mn></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mi>b</mi></mrow></msup></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mi>s</mi></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525214&type= 150.19866943 16.67933464 可见 <math id="M89"><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525216&type= 3.72533321 3.72533321 服从伽马分布 <math id="M90"><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525218&type= 11.00666714 3.55599999 。特别地， <math id="M91"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525220&type= 3.72533321 3.72533321 服从指数分布 <math id="M92"><mi mathvariant="normal">E</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525222&type= 11.34533310 3.72533321 。 例2 <math id="M93"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525224&type= 3.30199981 4.57200003 服从 <math id="M94"><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525226&type= 8.21266651 3.97933340 上的两点分布， <math id="M95"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525229&type= 22.60599899 4.65666676 ， <math id="M96"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525231&type= 29.20999908 4.65666676 ， <math id="M97"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525233&type= 15.15533447 3.64066648 ， <math id="M98"><mi>b</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℤ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525235&type= 10.07533360 3.38666677 ，计算 <math id="M99"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525237&type= 16.76399994 4.48733330 。 由定理1可知： <math id="M100"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>λ</mi><mi>p</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>p</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525239&type= 34.54399872 4.91066647 ,以及当 <math id="M101"><mi>b</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525241&type= 8.46666718 3.30199981 时， <math display="block" id="M102"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">d</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>λ</mi><mi>p</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub></mrow></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mi>λ</mi><mi>p</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi><mo>-</mo><mi>b</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>=</mo><mi>λ</mi><mi>p</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi><mo>-</mo><mi>b</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>!</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><mi>b</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mi>λ</mi><mi>p</mi><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi><mo>+</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>=</mo><mi>λ</mi><mi>p</mi><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup></mrow><mrow><mi>l</mi><mo>!</mo></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>b</mi></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><msup><mrow><mi>s</mi></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>s</mi></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>λ</mi><mi>s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfenced></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>λ</mi><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>b</mi></mrow></msup></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mi>s</mi></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>p</mi><mi>s</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>s</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525243&type= 143.84866333 48.68333054 可见 <math id="M103"><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525245&type= 3.72533321 3.72533321 服从伽马分布 <math id="M104"><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>λ</mi><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525247&type= 12.95399952 3.72533321 。特别地， <math id="M105"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525249&type= 3.72533321 3.72533321 服从指数分布 <math id="M106"><mi mathvariant="normal">E</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525252&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525251&type= 13.29266644 3.72533321 。 例3 <math id="M107"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525253&type= 3.30199981 4.57200003 服从 <math id="M108"><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525256&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525255&type= 22.35199928 3.55599999 上的均匀分布， <math id="M109"><mi>b</mi><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525258&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525257&type= 23.79133415 3.97933340 ，计算 <math id="M110"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525259&type= 18.28800011 4.48733330 。 将 <math id="M111"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo>…</mo><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525261&type= 50.20733261 7.62000036 代入（4）式，得 <math display="block" id="M112"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525266&type= 143.00199890 12.61533356 以上最内层级数的结果为 <math id="M113"><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525272&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525270&type= 16.59466553 9.48266697 的 <math id="M114"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525274&type= 3.38666677 3.72533321 倍，其中 <math id="M115"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525278&type= 3.38666677 3.72533321 为满足 <math id="M116"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525281&type= 41.40200043 4.48733330 ， <math id="M117"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525286&type= 34.12066650 3.81000018 的 <math id="M118"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525290&type= 1.18533337 2.79399991 元组（ <math id="M119"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525296&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525294&type= 18.79599953 4.14866638 ）的数量。因为 <math id="M120"><mi>h</mi><mo>≤</mo><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo><</mo><mi>b</mi><mo>≤</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525297&type= 28.19400024 3.30199981 ， <math id="M121"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525301&type= 3.38666677 3.72533321 等价于 <math id="M122"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525305&type= 34.12066650 3.81000018 的非负整数解个数。由组合计数原理可知 <math id="M123"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>+</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>h</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525309&type= 18.88066673 3.97933340 ，故级数 <math id="M124"><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525313&type= 39.28533173 12.61533356 等于 <math id="M125"><mfrac><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>+</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>h</mi></mrow></msubsup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525317&type= 16.59466553 8.72066689 ，因此， <math display="block" id="M126"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>+</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>h</mi></mrow></msubsup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525321&type= 57.99666595 8.72066689 例4 <math id="M127"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525325&type= 3.30199981 4.57200003 服从如下分布 <math id="M128"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525329&type= 1.43933344 2.37066674 <math id="M129"><mn>2</mn><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525333&type= 3.04799986 2.37066674 … <math id="M130"><mi>α</mi><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525337&type= 3.04799986 2.37066674 <math id="M131"><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525341&type= 11.76866722 3.04799986 … <math id="M132"><mi>n</mi><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525345&type= 2.96333337 2.37066674 <math id="M133"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525349&type= 1.60866666 3.13266659 <math id="M134"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525353&type= 1.60866666 3.13266659 … <math id="M135"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525357&type= 1.60866666 3.13266659 <math id="M136"><msub><mrow><mi>p</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525361&type= 6.09599972 3.89466667 … <math id="M137"><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525365&type= 2.70933342 3.89466667 其中 <math id="M138"><mi>α</mi><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo><mo>,</mo><mi>k</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>α</mi><mi>p</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mi>α</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525373&type= 70.18866730 8.46666718 。 <math id="M139"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525377&type= 1.86266661 2.79399991 满足 <math id="M140"><mo>∃</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>α</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525381&type= 34.12066650 4.06400013 ，使得 <math id="M141"><mi>b</mi><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525385&type= 11.59933376 3.72533321 。 由于 <math id="M142"><mi>k</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525389&type= 8.38199997 2.79399991 ，故此例可归于 <math id="M143"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525393&type= 9.90599918 4.57200003 的情况。由于 <math id="M144"><mi>b</mi><mo>≤</mo><mi>n</mi><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525397&type= 10.07533360 3.30199981 ， <math id="M145"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>α</mi><mo><</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525401&type= 36.15266418 4.23333359 ，由（5）式可得 <math display="block" id="M146"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>k</mi><mo>-</mo><mi>h</mi><mi>k</mi></mtd></mtr></mtable></mrow></munder><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mrow></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><mi>p</mi></mrow></mrow></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><mi>p</mi></mrow><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo/></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525405&type= 120.22666168 31.41133308 当 <math id="M147"><mi>l</mi><mo>></mo><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525409&type= 8.04333305 2.79399991 时， <math id="M148"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>≥</mo><mi>l</mi><mo>></mo><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525412&type= 40.30133438 4.14866638 ，即满足条件 <math id="M149"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525417&type= 41.23266602 4.48733330 ， <math id="M150"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525421&type= 34.12066650 3.81000018 的 <math id="M151"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525425&type= 1.18533337 2.79399991 元组 <math id="M152"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525429&type= 21.75933456 4.48733330 不存在；当 <math id="M153"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525433&type= 14.47799969 3.30199981 时，因为 <math id="M154"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>α</mi><mo><</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525437&type= 36.15266418 4.23333359 ，故满足以上条件的 <math id="M155"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525441&type= 21.75933456 4.48733330 必然存在，且所有这样的 <math id="M156"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525445&type= 21.75933456 4.48733330 恰好是 <math id="M157"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525449&type= 34.12066650 3.81000018 的所有正整数解，此时最内层级数的结果为 <math id="M158"><msup><mrow><mi>p</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525453&type= 6.18066692 3.89466667 的 <math id="M159"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525459&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525457&type= 3.38666677 3.72533321 倍，其中 <math id="M160"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525463&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525461&type= 3.38666677 3.72533321 为满足以上条件的 <math id="M161"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525465&type= 21.75933456 4.48733330 组数，则 <math id="M162"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525471&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525469&type= 3.38666677 3.72533321 等价于 <math id="M163"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525473&type= 34.12066650 3.81000018 的正整数解个数。由组合计数原理可知 <math id="M164"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525477&type= 15.66333389 3.97933340 ，故级数 <math id="M165"><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525481&type= 28.87133217 9.65199947 等于 <math id="M166"><msubsup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msup><mrow><mi>p</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525485&type= 14.30866623 4.82599974 ，因此， <math display="block" id="M167"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mrow></mrow><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>h</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msup><mrow><mi>p</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mrow><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></msubsup><msup><mrow><mi>p</mi></mrow><mrow><mi>l</mi></mrow></msup></mrow></mrow><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525491&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525489&type= 114.80799866 23.45266724 特别地，若 <math id="M168"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>≠</mo><mi>α</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525493&type= 16.67933464 3.97933340 ，则 <math id="M169"><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>=</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525497&type= 10.66799927 4.06400013 ，此时 <math id="M170"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525501&type= 42.07933426 5.16466665 。 例5 （独立同参数泊松过程的加权和） <math id="M171"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525507&type= 3.30199981 4.57200003 服从 <math id="M172"><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525513&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525510&type= 18.54199982 3.55599999 上的均匀分布， <math id="M173"><mi>b</mi><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525517&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525515&type= 23.19866753 3.97933340 ，计算 <math id="M174"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525519&type= 17.69533348 4.48733330 。 在例4中取 <math id="M175"><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525524&type= 8.38199997 2.79399991 ， <math id="M176"><mi>α</mi><mo>=</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525530&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525528&type= 15.24000072 2.79399991 以及 <math id="M177"><mi>p</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525532&type= 9.82133293 7.53533316 ，代入例4结论得： <math id="M178"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>b</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525538&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525536&type= 43.94200134 9.48266697 。 这是对文献［ 11 11 ］中定理2.1结论的修正。 例6 <math id="M179"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525540&type= 3.30199981 4.57200003 服从 <math id="M180"><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525547&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525544&type= 18.54199982 3.55599999 上的均匀分布，计算 <math id="M181"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525549&type= 21.84399986 4.48733330 和 <math id="M182"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525552&type= 21.84399986 4.48733330 。 由（5）式知，当 <math id="M183"><mi>b</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525555&type= 15.07066631 2.96333337 或 <math id="M184"><mi>b</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525559&type= 15.07066631 2.96333337 时， <math id="M185"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mrow></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525564&type= 67.98733521 11.51466751 ，故 <math display="block" id="M186"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>h</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msup><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo stretchy="false">[</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525569&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525567&type= 118.95666504 22.43666649 <math display="block" id="M187"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>⋯</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>h</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>h</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>h</mi></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow></mrow><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mfrac><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>l</mi></mrow></msubsup></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi></mrow></msup></mrow></mfrac></mrow><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mfrac><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow><mtext> </mtext><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msup><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>l</mi></mrow></msubsup></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msup><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>h</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>l</mi></mrow></msubsup></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>l</mi></mrow></msup></mrow></mfrac><mo>+</mo></mrow><mfrac><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo stretchy="false">)</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">(</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo> </mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">[</mo><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>n</mi></mrow></msup><mo stretchy="false">]</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525571&type= 120.39599609 78.06266785 这也是对文献［ 11 11 ］中定理2.1结论的修正。 例7 （独立泊松过程的加权和）令 <math id="M188"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfrac><mo>,</mo><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525575&type= 53.00133133 13.20799923 ，其中 <math id="M189"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525581&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525579&type= 25.82333374 4.14866638 ，计算 <math id="M190"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525584&type= 18.96533394 4.48733330 。 由（5）式得 <math display="block" id="M191"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfrac><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>h</mi></mrow></msub><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow></mrow><mo>,</mo><mi mathvariant="normal">若</mi><mo>∃</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">使</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">b</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><munder><mo movablelimits="false">∑</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo>=</mo><mi>b</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></mtd></mtr></mtable></mrow></munder><mrow><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>h</mi></mrow></msub><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac></mrow></mrow></mrow><mo>,</mo><mi mathvariant="normal">若</mi><mi mathvariant="normal">b</mi><mo>></mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">其</mi><mi mathvariant="normal">他</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525593&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525590&type= 118.36399841 48.09066772 记 <math id="M192"><mi>λ</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525600&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525596&type= 14.13933372 8.88999939 ，结合文献［ 12 12 ］中的命题（2.6），此时的 <math id="M193"><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo></mrow><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525608&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525604&type= 29.46399879 8.88999939 ，其中 <math id="M194"><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525618&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525613&type= 8.63599968 4.48733330 为参数为 <math id="M195"><msub><mrow><mi>λ</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525627&type= 2.45533323 3.72533321 的泊松过程，且各 <math id="M196"><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525637&type= 8.63599968 4.48733330 相互独立。因此（6）式是文献［ 12 12 ］中定理2.7的结论的另一种表达形式。 3　 结 语 1 本文计算了一类带正跳跃的复合泊松过程首达时的概率密度函数及首达时有限的概率，给出了一些具体例子的计算结果。文献［ 9 9 ~ 11 11 9~11 ］中的一些相关结论皆为本文相关结论的特例。从计算结果可以看出，本文讨论的复合泊松过程首达时的概率密度表达式较为复杂，一般来说是难以化简的级数，在实际应用中，若需要具体的数值结果，经常要用到计算机模拟。本文只讨论了各个 <math id="M197"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525646&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525643&type= 3.30199981 4.57200003 为非负随机变量并且满足有限离散分布的情况，后续研究还可以考虑 <math id="M198"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16525649&type= 3.30199981 4.57200003 满足无穷离散分布、连续分布或者不一定非负的情况。更进一步地，还可以考虑计算已知 Q 矩阵情况下一般的连续时间离散状态马氏过程的首达时概率密度。 参考文献 SHEPP L A . A first passage problem for the Wiener process ［J］. The Annals of Mathematical Statistics ， 1967 ， 38 （ 6 ）： 1912 - 1914 . DOI： 10.1214/aoms/1177698626 http://dx.doi.org/10.1214/aoms/1177698626 . A first passage problem for the Wiener process NOVIKOV A A . On stopping times for a Wiener process ［J］. Theory of Probability & Its Applications ， 1971 ， 16 （ 3 ）： 449 - 456 . DOI： 10.1137/1116049 http://dx.doi.org/10.1137/1116049 . On stopping times for a Wiener process ITO K ， MCKEAN H . Diffusion Processes and Their Sample Paths ［M］. Berlin ： Springer ， 1974 ： 25 - 33 . Diffusion Processes and Their Sample Paths KARATZAS I ， SHREVE S E . Brownian Motion and Stochastic Calculus ［M］. 2nd ed . New York， Berlin ： Springer ， 1991 ： 79 - 81 . Brownian Motion and Stochastic Calculus RICCIARDI L M ， SATO S . First-passage-time density and moments of the Ornstein-Uhlenbeck process ［J］. Journal of Applied Probability ， 1988 ， 25 （ 1 ）： 43 - 57 . DOI： 10.2307/3214232 http://dx.doi.org/10.2307/3214232 . First-passage-time density and moments of the Ornstein-Uhlenbeck process YI C . On the first passage time distribution of an Ornstein-Uhlenbeck process ［J］. Quantitative Finance ， 2010 ， 10 （ 9 ）： 957 - 960 . DOI： 10.1080/14697680903373684 http://dx.doi.org/10.1080/14697680903373684 . On the first passage time distribution of an Ornstein-Uhlenbeck process LIPTON A ， KAUSHANSKY V . On the First Hitting Time Density of an Ornstein-Uhlenbeck Process ［EB/OL］. ［ 2020-10-10 ］. https：//arxiv.org/pdf/1810.02390v2.pdf https://arxiv.org/pdf/1810.02390v2.pdf . 10.1080/14697688.2020.1713394 http://dx.doi.org/10.1080/14697688.2020.1713394 On the First Hitting Time Density of an Ornstein-Uhlenbeck Process LIPTON A ， KAUSHANSKY V . On the first hitting time density for a reducible diffusion process ［J］. Quantitative Finance ， 2020 ， 20 （ 5 ）： 723 - 743 . DOI： 10.1080/14697688.2020.1713394 http://dx.doi.org/10.1080/14697688.2020.1713394 . On the first hitting time density for a reducible diffusion process PESKIR G . The law of the hitting times to points by a stable Lévy process with no negative jumps ［J］. Electronic Communications in Probability ， 2008 ， 13 ： 653 - 659 . DOI： 10.1214/ecp.v13-1431 http://dx.doi.org/10.1214/ecp.v13-1431 . The law of the hitting times to points by a stable Lévy process with no negative jumps SENGAR A S ， MAHESHWARI A ， UPADHYE N S . Time-changed Poisson processes of order K ［J］. Stochastic Analysis and Applications ， 2020 ， 38 （ 1 ）： 124 - 148 . DOI： 10.1080/07362994.2019.1653198 http://dx.doi.org/10.1080/07362994.2019.1653198 . Time-changed Poisson processes of order K MAHESHWARI A ， ORSINGHER E ， SENGAR A S . Superposition of Time-Changed Poisson Processes and Their Hitting Times ［EB/OL］. ［ 2020-11-12 ］. https： //arxiv.org/abs/1909.13213 https://arxiv.org/abs/1909.13213 . Superposition of Time-Changed Poisson Processes and Their Hitting Times ZUO H ， SHEN Z H ， RANG G L . Hitting probabilities of weighted Poisson processes with different intensities and their subordinations ［J］. Acta Mathematica Scientia ， 2021 ， 41 （ 1 ）： 67 - 84 . DOI： 10.1007/s10473-021-0104-6 http://dx.doi.org/10.1007/s10473-021-0104-6 . Hitting probabilities of weighted Poisson processes with different intensities and their subordinations 10.14188/j.1671-8836.2021.0023 O211.62 A 1671-8836（2021）03-0256-07 陶秀丽,沈兆晖.一类复合泊松过程的首达时概率密度[J].武汉大学学报（理学版）,2021,67(3):256-262. DOI:10.14188/j.1671-8836.2021.0023 陶秀丽,沈兆晖.一类复合泊松过程的首达时概率密度[J].武汉大学学报（理学版）,2021,67(3):256-262. DOI:10.14188/j.1671-8836.2021.0023 TAO Xiuli,SHEN Zhaohui.On the First Passage Time Density for a Class of Compound Poisson Process [J]. J Wuhan Univ （ Nat Sci Ed ）,2021,67(3):256-262. DOI: 10.14188/j.1671-8836.2021.0023 http://dx.doi.org/10.14188/j.1671-8836.2021.0023 (Ch). TAO Xiuli,SHEN Zhaohui.On the First Passage Time Density for a Class of Compound Poisson Process [J].J Wuhan Univ（Nat Sci Ed）,2021,67(3):256-262. DOI:10.14188/j.1671-8836.2021.0023(Ch). 2021-01-26 2021-06-24 2021-06-11 武汉大学学报（理学版） 3 67