The Annals of Mathematical Statistics

The Probability Function of the Product of Two Normally Distributed Variables

Leo A. Aroian

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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.

Article information

Source
Ann. Math. Statist. Volume 18, Number 2 (1947), 265-271.

Dates
First available: 28 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aoms/1177730442

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aoms/1177730442

Mathematical Reviews number (MathSciNet)
MR21284

Zentralblatt MATH identifier
0041.45004

Citation

Aroian, Leo A. The Probability Function of the Product of Two Normally Distributed Variables. The Annals of Mathematical Statistics 18 (1947), no. 2, 265--271. doi:10.1214/aoms/1177730442. http://projecteuclid.org/euclid.aoms/1177730442.


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