Multiple of Chi-Squared Random Variable has Gamma Distribution

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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:

\(\ds \map \Pr {Y < k x}\) \(=\) \(\ds \frac 1 {\map \Gamma {n / 2} } \paren {\frac 1 {2 k} }^{n / 2} \int_0^{k x} t^{\paren {n / 2} - 1} e^{-\paren {1 / k} t / 2} \rd t\) Definition of Gamma Distribution
\(\ds \) \(=\) \(\ds \frac 1 {\map \Gamma {n / 2} } \paren {\frac 1 {2 k} }^{n / 2} \int_0^x k \paren {k u}^{\paren {n / 2} - 1} e^{-u / 2} \rd u\) substituting $t = k u$
\(\ds \) \(=\) \(\ds \frac {k^{n / 2} } {\map \Gamma {n / 2} \paren {2 k}^{n / 2} } \int_0^x u^{\paren {n / 2} - 1} e^{-u / 2} \rd u\)
\(\ds \) \(=\) \(\ds \frac 1 {2^{n / 2} \map \Gamma {n / 2} } \int_0^x u^{\paren {n / 2} - 1} e^{-u / 2} \rd u\)
\(\ds \) \(=\) \(\ds \map \Pr {X < x}\) Definition of Chi-Squared Distribution

$\blacksquare$