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A Characterization of the Normal andWeibull Distributions

Published online by Cambridge University Press:  20 November 2018

V. Seshadri
V. Seshadri*
Affiliation:
McGill University
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Let X and Y be two independent normal variates each distributed with zero mean and a common variance. Then the quotient X/Y has the Cauchy distribution symmetrical about the origin. Of particular interest in recent years has been the converse problem and examples of non-normal distributions with a Cauchy distribution for the quotient have been illustrated by Mauldon [9], Laha [2; 3; 4] and Steck [10].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 3 , June 1969 , pp. 257 - 260
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Kawata, T. and Sakamoto, H., On the characterization of the normal distribution by the independence of the sample mean and the sample variance. J. Math. Soc. Japan 1 (1949) 111–115.Google Scholar
2. Laha, R. G., An example of a non-normal distribution where the quotient follows the Cauchy law. Proc. Nat. Acad. Sci. U.S.A. 44 (1958) 222–223.Google Scholar
3. Laha, R. G., On the laws of Cauchy and Gauss. Ann. Math. Stat. 30 (1959) 1165–1174.Google Scholar
4. Laha, R. G., On a class of distribution functions where the quotient follows the Cauchy law. Trans. Amer. Math. Soc. 93 (1959) 205–215.Google Scholar
5. Laha, R. G., Lukacs, E., and Newman, M., On the independence of a sample central moment and the sample mean. Ann. Math. Stat. 31 (1960) 1028–1033.Google Scholar
6. Lukacs, E., A characterization of the normal distribution. Ann. Math. Stat. 13 (1942) 91–93.Google Scholar
7. Lukacs, E., A characterization of the gamma distribution. Ann. Math. Stat. 26 (1955) 319–324.Google Scholar
8. Lukacs, E., The stochastic independence of symmetric and homogeneous linear and quadratic statistics. Ann. Math. Stat. 23 (1952) 442–449.Google Scholar
9. Mauldon, J. G., Characterizing properties of statistical distributions. Quart. J. Math. Oxford Ser. (2) 7 (1956) 155–160.Google Scholar
10. Steck, G. P., A uniqueness property not enjoyed by the normal distribution. Ann. Math. Stat. 31 (1958) 1028–1033.Google Scholar