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 $n \in \mathbb{Z}^*$ degrees of freedom is a special case of the gamma distribution with parameters $\alpha = \frac{n}{2}$, $\beta = \frac{1}{2}$ and support $t < \frac{1}{2}$.

Proof: The moment-generating function of a gamma random variable $X \sim \mathrm{Gam}(\alpha, \beta)$ is

\[\label{eq:gam-mgf} M_X(t) = (1-\frac{t}{\beta})^{-\alpha}, \; t < \beta, \; \beta > 0 \; .\]

The moment-generating function of the a chi-squared random variable $Y \sim \chi^2_{n}$ with $n$ degrees of freedom is

\[\label{eq:chi2-mgf} M_Y(t) = (1-2t)^{-n/2}, \; t < \frac{1}{2} \; .\]

If there is a random variable $X \sim \mathrm{Gam}(n/2, 1/2)$, then

\[\label{eq:gam-chi2-mgf} M_X(t) = (1-\frac{t}{\frac{1}{2}})^{-(n/2)} = (1-2t)^{-n/2} = M_Y(t) \quad \text{for} \quad t < \frac{1}{2}\]

“If the moment-generating function exists for $t$ in an open interval containing zero, it uniquely determines the probability distribution.” (Rice, 2007, p. 155) Both moment-generating functions \eqref{eq:gam-mgf} and \eqref{eq:chi2-mgf} satisfy this condition. Specifically, $M_X(t)$ is defined for $t < \frac{1}{2}$, and for $M_Y(t)$, we have $t < \beta$ with $\beta > 0$; both of these intervals contain 0.

By the uniqueness of the moment-generating function, the chi-squared distribution with $n$ degrees of freedom is a special case of the gamma distribution with parameters $\alpha = \frac{n}{2}$ and $\beta = \frac{1}{2}$ and support $t < \frac{1}{2}$. This completes the proof.

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Metadata: ID: P521 | shortcut: chi2-gam2 | author: natrium-256 | date: 2026-01-02, 16:47.