Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-03T01:21:22.997Z Has data issue: false hasContentIssue false
  • English
  • Français

A Family of Distributions with the same Ratio Property as Normal Distribution

Published online by Cambridge University Press:  20 November 2018

Charles Fox
Charles Fox*
Affiliation:
McGill University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If U and V are independent random variables, both drawn from the Normal distribution, then it is known that the distribution of U/V follows the Cauchy law, i. e. has frequency function 1/π(1+x2). Conversely if U and V are independently drawn from the same distribution and U/V is known to follow the Cauchy law of distribution must U and V be necessarily drawn from a Normal distribution? This question has been considered by several authors ([3], [4], [5], [7]) who have obtained several examples to show that the ratio property above is not confined to the Normal distribution.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 8 , Issue 5 , October 1965 , pp. 631 - 636
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Bromwich, T. J. I' a., An Introduction to the Theory of Infinite Series. MacMillan, London, 1908.Google Scholar
2. Fox, C., The Solution of an Integral Equation. Canadian Journal of Mathematics, vol. 16, (1964) pp. 578-586.Google Scholar
3. Laha, R. G., An example of a non-normal distribution where the quotient follows the Cauchy law. The National Academy of Sciences (U. S. A.)44 (1958) pp. 222-223.Google Scholar
4. Laha, R. G., On the laws of Cauchy and Gauss. Annals of Mathematical Statistics, 30 (1959) pp. 1165-1174.Google Scholar
5. Mauldon, J. G., Characterizing properties of Statistical Distributions. Quarterly Journal of Mathematics (Oxford) series 2, vol.7 (1956) pp. 155-160.Google Scholar
6. Sneddon, I.N., Fourier Transforms. McGraw Hill, New York, 1951.Google Scholar
7. Steck, G. P., A uniqueness property not enjoyed by the Normal Distribution. Annals of Mathematical Statistics, 29 (1958) pp. 604-606.Google Scholar
8. Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis. Cambridge University Press, 1915.Google Scholar
9. Watson, G.N., Theory of Bessel Functions. Cambridge University Press, 1922.Google Scholar
10. Cramér, H., Mathematical Methods of Statistics. Princeton University Press, 1946.Google Scholar