Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Chi-squared distribution ▷ Special case of gamma distribution

Theorem: The chi-squared distribution with $k$ degrees of freedom is a special case of the gamma distribution with shape $\frac{k}{2}$ and rate $\frac{1}{2}$:

\[\label{eq:chi2-gam} X \sim \mathrm{Gam}\left( \frac{k}{2}, \frac{1}{2} \right) \Rightarrow X \sim \chi^{2}(k) \; .\]

Proof: The probability density function of the gamma distribution for $x > 0$, where $\alpha$ is the shape parameter and $\beta$ is the rate parameter, is as follows:

\[\label{eq:gam-pdf} \mathrm{Gam}(x; \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \, x^{\alpha-1} \, e^{-\beta x}\]

If we let $\alpha = k/2$ and $\beta = 1/2$, we obtain

\[\label{eq:gam-pdf-chi2} \mathrm{Gam}\left(x; \frac{k}{2}, \frac{1}{2}\right) = \frac{x^{k/2-1} \, e^{-x/2}}{\Gamma(k/2) 2^{k/2}} = \frac{1}{2^{k/2} \Gamma(k/2)} \, x^{k/2-1} \, e^{-x/2}\]

which is equivalent to the probability density function of the chi-squared distribution.

∎
Sources:

Metadata: ID: P174 | shortcut: chi2-gam | author: kjpetrykowski | date: 2020-10-12, 22:15.