A remark on the distribution of products of independent normal random variables

Authors

DOI:

https://doi.org/10.5604/01.3001.0014.7861

Keywords:

normal distribution, Maijer G-functions

Abstract

We present a proof of the explicit formula of the probability density function of the product of normally distributed independent random variables using the multiplicative convolution formula for Meijer G functions.

Downloads

Download data is not yet available.

Bateman H, Erdélyi A. Higher transcendental functions. Vol. 1. New York: McGraw-Hill; 1953.   Google Scholar

Beals R, Szmigielski J. Meijer G-functions: a gentle introduction. Notices of the American Mathematical Society. 2013;60(7):866–872. doi: http://dx.doi.org/10.1090/notimanid1016.   Google Scholar

Carter BD, Springer MD. The distribution of products, quotients and powers of independent H-function variates. SIAM Journal on Applied Mathematics. 1977;33(4):542–558. doi: https://doi.org/10.1137/0133036.   Google Scholar

Craig CC. On the frequency function of XY. The Annals of Mathematical Statistics. 1936;7(1):1–15.   Google Scholar

Craig CC. On frequency distributions of the quotient and of the product of two statistical variables. The American Mathematical Monthly. 1942;49(1):24–32. doi: https://doi.org/10.1080/00029890.1942.11991175.   Google Scholar

Fisz M. Probability and Mathematical Statistics. Delhi: S. Chand; 1961.   Google Scholar

Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. 7th ed. Amsterdam: Academic Press; Elsevier; 2007.   Google Scholar

Springer MD. The Algebra of Random Variables. Wiley Series in Probability and Mathematical Statistics. New York–Chichester–Brisbane: John Wiley & Sons; 1979.   Google Scholar

Springer MD, Thompson WE. The distribution of products of independent random variables. SIAM Journal on Applied Mathematics. 1966;14(3):511–526. doi: https://doi.org/10.1137/0114046.   Google Scholar

Springer MD, Thompson WE. The distribution of products of beta, gamma and Gaussian random variables. SIAM Journal on Applied Mathematics. 1970;18(4):721–737. doi: https://doi.org/10.1137/0118065.   Google Scholar

Wishart J, Barlett MS. The distribution of second order moment statistics in a normal system. Mathematical Proceedings of the Cambridge Philosophical Society. 1932;28(4):455–459. doi: https://doi.org/10.1017/S0305004100010690.   Google Scholar

Downloads

Published

2021-03-10

How to Cite

Szczepański, J. (2021). A remark on the distribution of products of independent normal random variables. Science, Technology and Innovation, 10(3), 30–37. https://doi.org/10.5604/01.3001.0014.7861

Issue

Section

Original articles