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Chi Squared Distribution

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Distribution

If _ ~X_1, ... ~X_~k _ are independent standard normal random variables, then _ ~Y _ #:= _ ~X_1^2 + ... + ~X_~k^2 _ has a #~{chi-squared distribution}, with ~k #~{degrees of freedom} where

f ( ~y ) _ = _ fract{~y ^{~k/2 - 1} e ^{-~y/2} , 2 ^{~k/2} &Gamma.( ~k/2 )}

where &Gamma.(&alpha.) is the gamma function

We write _ ~Y ~ &chi.^2 (~k). Note the following particular cases:

~Y ~ &chi.^2 (1) _ _ _ => _ _ _ f ( ~y ) _ = _ fract{ e ^{-~y/2} , &sqrt.${2 &pi. ~y}}

~Y ~ &chi.^2 (2) _ _ _ => _ _ _ f ( ~y ) _ = _ fract{ e ^{-~y/2} , 2 }

In fact the Chi-squared distribution is just a special case of the Gamma distribution. _ If _ ~Y ~ &chi.^2 (~k) , _ then ~Y ~ &Gamma. ( ~k/2 , 1/2 )

Expectation

If _ ~Y ~ &chi.^2 (1) , _ then

E ~Y _ = _ fract{1, &sqrt.$2 &Gamma.( 1/2 )} int{,0,&infty.,,} ~y^{ 1/2 } e ^{-~y/2} _ d~y

but _ ~{&integ.}__0^^{&infty.} ~y^{ 1/2 } e ^{-~y/2} d~y _ = _ 2 &sqrt.$2 ~{&integ.}__0^^{&infty.} ~u^{ 1/2 } e ^{-~u} d~u _ = _ 2 &sqrt.$2 &Gamma. ( 3/2 ) _ = _ 2 &sqrt.$2 1/2 &Gamma. ( 1/2 )

So if _ ~Y ~ &chi.^2 (1) , _ then _ E ( ~Y ) _ = _ 1.