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On Random Variables which have the same Distribution as their Reciprocals

Published online by Cambridge University Press:  20 November 2018

V. Seshadri
V. Seshadri*
Affiliation:
McGill University
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The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 8 , Issue 6 , December 1965 , pp. 819 - 824
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Bose, S. S., (1935), On the distribution of the ratio of variances of two samples drawn from a given normal bivariate correlated population, Sankhya 2, p. 65.Google Scholar
2. Finney, D.J., (1938), The distribution of the ratio of estimates of the two variances in a sample from a normal bivariate population, Biometrika, 30, p. 190.Google Scholar
3. Murty, V.N., (1950), Distribution of the quotient of maximum values in samples from a rectangular distribution, Journal of the American Statistical Association, 50, p. 1136.Google Scholar