The Probability Function of the Product of Two Normally Distributed Variables

Abstract

Let $x$ and $y$ follow a normal bivariate probability function with means $\bar X, \bar Y$, standard deviations $\sigma_1, \sigma_2$, 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 $z = xy/\sigma_1\sigma_2$ 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 $z, \rho_1, \rho_2$, 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 $\rho_1$ and $\rho_2$ large, as Craig points out, the series expansion converges very slowly. Even for $\rho_1$ and $\rho_2$ as small as 2, the expansion is unwieldy. We shall show that as $\rho_1$ and $\rho_2 \rightarrow \infty$, 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.