## Chi-Squared Distribution

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
 (4)

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
 (5) (6) (7) (8)

so
 (9) (10) (11) (12)

The th Moment about zero for a distribution with Degrees of Freedom is
 (13)

and the moments about the Mean are
 (14) (15) (16)

The th Cumulant is
 (17)

 (18)

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 .

 (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
 (34) (35) (36) (37)

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.