Expand/Contract All

Gamma Distribution

Page Contents

Distribution

A random varuiable ~Y has the #~{gamma distribution} with parameters &alpha. ("shape") and &beta. ("rate") if it has the density function:

f ( ~y ; &alpha. , &beta. ) _ = _ fract{&beta.^{&alpha.} ~y^{&alpha. - 1} e^{-&beta.~y} , &Gamma.( &alpha. )}

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

We write _ ~y _ ~ _ &Gamma. ( &alpha. , &beta. ).

The distribution function is therefore:

F ( ~y ; &alpha. , &beta. ) _ = _ int{,0,~y,}fract{ _ &beta.^{&alpha.} ~x^{&alpha. - 1} e^{-&beta.~x} , &Gamma.( &alpha. )} d~x

but note

int{,0,~y,} &beta.^{&alpha.} ~x^{&alpha. - 1} e^{-&beta.~x} d~x _ = _ int{,0,&beta.~y,} ~t^{&alpha. - 1} e^{-~t} d~t

using the substitution _ ~t = &beta.~x . _ [ d~x = d~t ./ &beta. _ etc. ]