The Probability Function of the Product of Two Normally Distributed Variables

Let x and y follow a normal bivariate probability function with means$\bar X, \bar Y$, standard deviations σ1, σ2, respectively, r the coefficient of correlation, and$\rho_1 = \bar X/\sigma_1, \rho_2 = \bar Y/\sigma_2$. Professor C. C. Craig [1] has found the probability function of z = xy/σ1σ2in 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 ρ1and ρ2large, as Craig points out, the series expansion converges very slowly. Even for ρ1and ρ2as small as 2, the expansion is unwieldy. We shall show that as ρ1and 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.