Chi-Squared Distribution

If  have normal independent distributions with mean 0 and variance 1, then

(1)

is distributed as  with  degrees of freedom. This makes a  distribution a gamma distribution with  and , where  is the number of degrees of freedom.

More generally, if  are independently distributed according to a  distribution with , , ...,  degrees of freedom, then

(2)

is distributed according to  with  degrees of freedom.

The probability density function for the  distribution with  degrees of freedom is given by

(3)

for , where  is a gamma function. The cumulative distribution function is then

			(4)
			(5)
			(6)
			(7)

where  is an incomplete gamma function and  is a regularized gamma function.

The chi-squared distribution is implemented in the Wolfram Language as ChiSquareDistribution[n].

For ,  is monotonic decreasing, but for , it has a maximum at

(8)

where

(9)

The th raw moment for a distribution with  degrees of freedom is

			(10)
			(11)

giving the first few as

			(12)
			(13)
			(14)
			(15)

The th central moment is given by

(16)

where  is a confluent hypergeometric function of the second kind, giving the first few as

			(17)
			(18)
			(19)
			(20)

The cumulants can be found via the characteristic function

			(21)
			(22)

Taking the natural logarithm of both sides gives

(23)

But this is simply a Mercator series

(24)

with , so from the definition of cumulants, it follows that

(25)

giving the result

(26)

The first few are therefore

			(27)
			(28)
			(29)
			(30)

The moment-generating function of the  distribution is

			(31)
			(32)
			(33)
			(34)
			(35)

so

			(36)
			(37)
			(38)
			(39)
			(40)
			(41)

If the mean is not equal to zero, a more general distribution known as the noncentral chi-squared distribution results. In particular, if  are independent variates with a normal distribution having means  and variances  for , ..., , then

(42)

obeys a gamma distribution with , i.e.,

(43)

where .

REFERENCES:

Abramowitz, M. and Stegun, I. 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.

Kenney, J. F. and Keeping, E. S. "The Chi-Square Distribution." §5.3 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 98-100, 1951.

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.