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June, 1958 A Uniqueness Property Not Enjoyed by the Normal Distribution
George P. Steck
Ann. Math. Statist. 29(2): 604-606 (June, 1958). DOI: 10.1214/aoms/1177706642


It is well known that if $X$ and $Y$ (or $1/X$ and $1/Y$) are independently normally distributed with mean zero and variance $\sigma^2,$ then $X/Y$ has a Cauchy distribution. It is the purpose of this note to show that the converse statement is not true. That is, the fact that the ratio of two independent, identically distributed, random variables $X$ and $Y$ follows a Cauchy distribution is not sufficient to imply that $X$ and $Y$ (or $1/X$ and $1/Y$) are normally distributed. This will be shown by exhibiting several counterexamples.


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George P. Steck. "A Uniqueness Property Not Enjoyed by the Normal Distribution." Ann. Math. Statist. 29 (2) 604 - 606, June, 1958.


Published: June, 1958
First available in Project Euclid: 27 April 2007

zbMATH: 0086.34203
MathSciNet: MR98442
Digital Object Identifier: 10.1214/aoms/1177706642

Rights: Copyright © 1958 Institute of Mathematical Statistics

Vol.29 • No. 2 • June, 1958
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