Let（Ω, B, P） be a probability space. Consider a random Dirichlet series of two variables where X,m, n,（ω）are random variables in（Ω, B, P） and λ,m, μ,n, satisfy the conditions 0,<,λ,1,<,λ,2,<,…,<,λ,m,↑+∞, 0,<,μ,1,<,μ,2,<,…,<,μ,n,↑+∞ Let D denote the largest open set in which the series is absolutely convergent and △ denote the largest connected open set in which the random analytic function represented by the series has analytic extention. When X,m, n,（ω） are independent symmetric random variables, we have obtained D=△ almost surely. This extends J. P. Kahane’s result in the ease of one variable.