On the computation of a bivariate 𝑡-distribution

The cumulative bivariate t t -distribution associated with random variables T 1 = X 1 / ( S / k ) 1 / 2 {T_1} = {X_1}/{(S/k)^{1/2}} , T 2 = X 2 / ( S / k ) 1 / 2 {T_2} = {X_2}/{(S/k)^{1/2}} is considered where X 1 {X_1} , X 2 {X_2} are bivariate normal with correlation coefficient ρ \rho and S S is an independent χ 2 {\chi ^2} random variable with k k degrees of freedom. Representations in terms of series and simple, one-dimensional quadratures are presented together with efficient computational procedures for the special functions used in numerical evaluation.