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 
.
