📂Probability Distribution

The Relationship Between the Gamma Distribution and the Chi-Squared Distribution

Theorem

$$\Gamma \left( { r \over 2 } , 2 \right) \iff \chi ^2 (r)$$

Description

The gamma distribution and the chi-square distribution have the following properties.

Proof

Strategy: It is shown that the moment-generating functions of the two distributions can be represented in the same form.

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The moment-generating function of the chi-square 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 $m_{2}(t) = (1-\theta t)^{-k}$. By substituting $\displaystyle k = {r \over 2}$ and $\theta = 2$ into the moment-generating function of the gamma distribution, we get $$m_{2}(t) = (1-\theta t)^{-k} = (1- 2t)^{- {r \over 2} } =m_{1}(t)$$

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