The Probability Function of the Product of Two Normally Distributed Variables

Let x and y follow a normal bivariate probability function with means X, Y, standard deviations σ₁, σ₂, respectively, r the coefficient of correlation, and ρ₁ = X/σ₁, ρ₂ = Y/σ₂. Professor C. C. Craig [1] has found the probability function of z = xy/σ₁σ₂ in closed form as the difference of two integrals. For purposes of numerical computation he has expanded this result in an infinite series involving powers of z, ρ₁, ρ₂, and Bessel functions of a certain type; in addition, he has determined the moments, semin-variants, and the moment generating function of z. However, for ρ₁ and ρ₂ large, as Craig points out, the series expansion converges very slowly. Even for ρ₁ and ρ₂ as small as 2, the expansion is unwieldy. We shall show that as ρ₁ and ρ₂ → ∞, the probability function of z approaches a normal curve and in case $r = 0$ the Type III function and the Gram-Charlier Type A series are excellent approximations to the z distribution in the proper region. Numerical integration provides a substitute for the infinite series wherever the exact values of the probability function of z are needed. Some extensions of the main theorem are given in section 5 and a practical problem involving the probability function of z is solved.