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(1) |
(Kenney and Keeping 1951). This general distribution includes the Chi-Squared Distribution and Student's t-Distribution as special cases. Let and be Independent Unbiased Estimators of the Variance of a Normally Distributed variate. Define
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(2) |
Then let
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(3) |
so that is a ratio of Chi-Squared variates
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(4) |
which makes it a ratio of Gamma Distribution variates,
which is itself a Beta Prime Distribution variate,
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(5) |
giving
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(6) |
The Mean is
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(7) |
and the Mode is
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(8) |
See also Beta Distribution, Beta Prime Distribution, Chi-Squared Distribution, Gamma Distribution,
Normal Distribution, Student's t-Distribution
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 180-181, 1951.
© 1996-9 Eric W. Weisstein
1999-05-26