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.

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