Inverse-gamma Distribution - Derivation From Gamma Distribution

Derivation From Gamma Distribution

The pdf of the gamma distribution is

and define the transformation then the resulting transformation is


f_Y(y) = f_X \left( g^{-1}(y) \right) \left| \frac{d}{dy} g^{-1}(y) \right|

=
\frac{1}{\theta^k \Gamma(k)}
\left( \frac{1}{y}
\right)^{k-1}
\exp \left( \frac{-1}{\theta y} \right)
\frac{1}{y^2}

=
\frac{1}{\theta^k \Gamma(k)}
\left( \frac{1}{y}
\right)^{k+1}
\exp \left( \frac{-1}{\theta y} \right)

=
\frac{1}{\theta^k \Gamma(k)}
y^{-k-1}
\exp \left( \frac{-1}{\theta y} \right).

Replacing with ; with ; and with results in the inverse-gamma pdf shown above


f(x)
=
\frac{\beta^\alpha}{\Gamma(\alpha)}
x^{-\alpha-1}
\exp \left( \frac{-\beta}{x} \right).

Read more about this topic:  Inverse-gamma Distribution

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