Relationship Between Gamma Distribution and Chi-Squared Distribution
Theorem
$$ \Gamma \left( { r \over 2 } , 2 \right) \iff \chi ^2 (r) $$
Explanation
The gamma distribution and the chi-squared distribution have the property shown above.
Proof
Strategy: We show that the moment generating functions of the two distributions can be expressed in the same form.
The moment generating function of the chi-squared distribution $\chi ^2 (r)$ is $\displaystyle m_{1}(t) = (1- 2t)^{- {r \over 2} }$, and the moment generating function of the gamma distribution $\Gamma (k, \theta)$ is $m_{2}(t) = (1-\theta t)^{-k}$. Substituting $\displaystyle k = {r \over 2}$ and $\theta = 2$ into the moment generating function of the gamma distribution gives $$ m_{2}(t) = (1-\theta t)^{-k} = (1- 2t)^{- {r \over 2} } =m_{1}(t) $$
■