Chi-squared Distribution

The χ² distribution is a non-negative continuous probability distribution function.

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Description

Given k normally distributed random variables, the sum of their squares follows a χ² distribution with k degrees of freedom.

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

for any x ∈ [0,∞), and where Γ is the gamma function.

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Moments

The expected value is given as E[X] = k.

Variance is given as Var[X] = 2k.

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Usage

Sum of Squared Residuals

The most common normal variable for which the sum of squares is taken is that of residuals.

Further note that the ratio of two χ² random variables is itself a random variable that follows the F distribution.

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CategoryRicottone