= Chi-squared Distribution =

The '''χ^2^ distribution''' is a non-negative continuous probability distribution function.

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== Description ==

Given ''k'' [[Analysis/NormalDistribution|normally distributed]] random variables, the sum of their squares follows a χ^2^ distribution with ''k'' [[Statistics/DegreesOfFreedom|degrees of freedom]].

This distribution is a special case of the [[Analysis/GammaDistribution|Gamma distribution]], with ''α=k/2'' and (scale) ''β = 2''. It follows that the distribution is given as:

{{attachment:chi.svg}}

for any ''x ∈ [0,∞)'', and where ''Γ'' is the [[Analysis/GammaFunction|gamma function]].

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== Moments ==

The [[Analysis/ExpectedValue|expected value]] is given as ''E[X] = k''.

[[Analysis/Variance|Variance]] is given as ''Var[X] = 2k''.

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== Usage ==



=== Sum of Squared Residuals ===

The most common [[Analysis/NormalDistribution|normal]] variable for which the sum of squares is taken is that of [[Statistics/Residuals|residuals]].

Further note that the ratio of two χ^2^ random variables is itself a random variable that follows the [[Analysis/FDistribution|F distribution]].



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