Multiple of Chi-Squared Random Variable has Gamma Distribution

Theorem
Let $n$ be a strictly positive integer.

Let $k > 0$ be a real number.

Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.

Then:


 * $k X \sim \map \Gamma {\dfrac n 2, \dfrac 1 {2 k}}$

where $\map \Gamma {\dfrac n 2, \dfrac 1 {2 k}}$ is the gamma distribution with parameters $\dfrac n 2$ and $\dfrac 1 {2 k}$.

Proof
Let:


 * $Y \sim \map \Gamma {\dfrac n 2, \dfrac 1 {2 k}}$

We aim to show that:


 * $\map \Pr {Y < k x} = \map \Pr {X < x}$

for all real $x \ge 0$.

We have: