Mean of Random Sample from Chi-Squared Distribution has Gamma Distribution

Theorem

Let $n$ be a strictly positive integer.

Let $X_1, X_2, \ldots, X_k$ form a random sample of size $k$ from the chi-squared distribution with $n$ degrees of freedom.

Then:

$\ds \overline X = \frac 1 k \sum_{i \mathop = 1}^k X_i \sim \map \Gamma {\frac {n k} 2, \frac k 2}$

where $\map \Gamma {\dfrac {n k} 2, \dfrac k 2}$ is the gamma distribution with parameters $\dfrac {n k} 2$ and $\dfrac k 2$.

Proof

By Sum of Chi-Squared Random Variables, we have:

$\ds \sum_{i \mathop = 1}^k X_i \sim \chi^2_{n k}$

By Multiple of Chi-Squared Random Variable has Gamma Distribution, we then have:

$\ds \frac 1 k \sum_{i \mathop = 1}^k X_i \sim \map \Gamma {\frac {n k} 2, \frac 1 {\frac 2 k} } = \map \Gamma {\frac {n k} 2, \frac k 2}$

$\blacksquare$