Sum of Chi-Squared Random Variables

Theorem

Let $n_1, n_2, \ldots, n_k$ be strictly positive integers which sum to $N$.

Let $X_i \sim {\chi^2}_{n_i}$ for $1 \le i \le k$, where ${\chi^2}_{n_i}$ is the chi-squared distribution with $n_i$ degrees of freedom.

Then:

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

Proof

Let $Y \sim {\chi^2}_N$.

By Moment Generating Function of Chi-Squared Distribution, the moment generating function of $X_i$ is given by:

$\map {M_{X_i} } t = \paren {1 - 2 t}^{-n_i / 2}$

Similarly, the moment generating function of $Y$ is given by:

$\map {M_Y} t = \paren {1 - 2 t}^{-N / 2}$

By Moment Generating Function of Linear Combination of Independent Random Variables, the moment generating function of $X$ is given by:

$\ds \map {M_X} t = \prod_{i \mathop = 1}^k \map {M_{X_i} } t$

We aim to show that:

$\map {M_X} t = \map {M_Y} t$

By Moment Generating Function is Unique, this ensures $X = Y$.

We have:

\(\ds \map {M_X} t\)

\(=\)

\(\ds \prod_{i \mathop = 1}^k \paren {1 - 2 t}^{-n_i / 2}\)

\(\ds \)

\(=\)

\(\ds \paren {1 - 2 t}^{-\paren {n_1 + n_2 + \ldots + n_k} / 2}\)

\(\ds \)

\(=\)

\(\ds \paren {1 - 2 t}^{-N / 2}\)

\(\ds \)

\(=\)

\(\ds \map {M_Y} t\)

$\blacksquare$