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September, 1948 Some Applications of the Mellin Transform in Statistics
Benjamin Epstein
Ann. Math. Statist. 19(3): 370-379 (September, 1948). DOI: 10.1214/aoms/1177730201


It is well known that the Fourier transform is a powerful analytical tool in studying the distribution of sums of independent random variables. In this paper it is pointed out that the Mellin transform is a natural analytical tool to use in studying the distribution of products and quotients of independent random variables. Formulae are given for determining the probability density functions of the product and the quotient $\frac{\xi}{\eta}$, where $\xi$ and $\eta$ are independent positive random variables with p.d.f.'s $f(x)$ and $g(y)$, in terms of the Mellin transforms $F(s) = \int_0^\infty f(x) x^{s-1} dx$ and $G(s) = \int_0^\infty g(y)y^{s-1} dy$. An extension of the transform technique to random variables which are not everywhere positive is given. A number of examples including Student's $t$-distribution and Snedecor's $F$-distribution are worked out by the technique of this paper.


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Benjamin Epstein. "Some Applications of the Mellin Transform in Statistics." Ann. Math. Statist. 19 (3) 370 - 379, September, 1948.


Published: September, 1948
First available in Project Euclid: 28 April 2007

zbMATH: 0032.29203
MathSciNet: MR29128
Digital Object Identifier: 10.1214/aoms/1177730201

Rights: Copyright © 1948 Institute of Mathematical Statistics

Vol.19 • No. 3 • September, 1948
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