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 .

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