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:


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

Proof
Let $Y \sim \chi^2_N$.

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


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

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


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

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


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

We aim to show that:


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

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

We have: