A distribution is a Gamma Distribution with
and
, where is the
number of Degrees of Freedom. If have Normal
Independent distributions with Mean 0 and Variance 1, then
|
(1) |
is distributed as with Degrees of Freedom. If are independently
distributed according to a distribution with , , ..., Degrees of Freedom, then
|
(2) |
is distributed according to with
Degrees of Freedom.
|
(3) |
The cumulative distribution function is then
where is a Regularized Gamma Function. The Confidence Intervals can be
found by finding the value of for which equals a given value. The Moment-Generating Function of the
distribution is
so
The th Moment about zero for a distribution with Degrees of Freedom is
|
(13) |
and the moments about the Mean are
The th Cumulant is
|
(17) |
The Moment-Generating Function is
As ,
|
(19) |
so for large ,
|
(20) |
is approximately a Gaussian Distribution with Mean and Variance . Fisher
showed that
|
(21) |
is an improved estimate for moderate . Wilson and Hilferty showed that
|
(22) |
is a nearly Gaussian Distribution with Mean and Variance
.
In a Gaussian Distribution,
|
(23) |
let
|
(24) |
Then
|
(25) |
so
|
(26) |
But
|
(27) |
so
|
(28) |
This is a distribution with , since
|
(29) |
If are independent variates with a Normal Distribution having Means and
Variances for , ..., , then
|
(30) |
is a Gamma Distribution variate with ,
|
(31) |
The noncentral chi-squared distribution is given by
|
(32) |
where
|
(33) |
is the Confluent Hypergeometric Limit Function and is the Gamma Function. The Mean,
Variance, Skewness, and Kurtosis are
See also Chi Distribution, Snedecor's F-Distribution
References
Abramowitz, M. and Stegun, C. A. (Eds.).
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 940-943, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 535, 1987.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
``Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function.'' §6.2 in
Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 209-214, 1992.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 115-116, 1992.
© 1996-9 Eric W. Weisstein
1999-05-26