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 ₁, ₂, respectively, r the coefficient of correlation, and ₁ = X/₁, ₂ = Y/₂. Professor C. C. Craig 1 has found the probability function of z = xy/₁₂ 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, ₁, ₂, 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 ₁ and ₂ large, as Craig points out, the series expansion converges very slowly. Even for ₁ and ₂ as small as 2, the expansion is unwieldy. We shall show that as ₁ and ₂, 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.