Chi-Squared Distribution

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

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

is distributed according to with degrees of freedom.

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

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

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

where

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

giving the first few as

The th central moment is given by

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

The cumulants can be found via the characteristic function

Taking the natural logarithm of both sides gives

But this is simply a Mercator series

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

giving the result

The first few are therefore

The moment-generating function of the distribution is

so

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

obeys a gamma distribution with , i.e.,

where .