If have normal independent distributions with mean 0 and variance 1, then
(1)
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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)
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is distributed according to with degrees of freedom.
The probability density function for the distribution with degrees of freedom is given by
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for , where is a gamma function. The cumulative distribution function is then
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(6)
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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)
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where
(9)
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The th raw moment for a distribution with degrees of freedom is
(10)
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giving the first few as
(12)
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The th central moment is given by
(16)
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where is a confluent hypergeometric function of the second kind, giving the first few as
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(19)
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The cumulants can be found via the characteristic function
(21)
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Taking the natural logarithm of both sides gives
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But this is simply a Mercator series
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with , so from the definition of cumulants, it follows that
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giving the result
(26)
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The first few are therefore
(27)
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The moment-generating function of the distribution is
(31)
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(34)
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so
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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)
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obeys a gamma distribution with , i.e.,
(43)
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where .