Chi-squared distribution

Definition

Let $k \ge 1$ be an integer and let $Z_1, \dots, Z_k$ be i.i.d. standard normal random variables. The chi-squared distribution with $k$ degrees of freedom, written $\chi^2_k$, is the law of $$Q = Z_1^2 + Z_2^2 + \dots + Z_k^2.$$

Remarks

It has the density $f(x) = \frac{x^{k/2 - 1} e^{-x/2}}{2^{k/2}\, \Gamma(k/2)}$ for $x > 0$, and $f(x) = 0$ for $x \le 0$. So $\chi^2_k$ is the gamma distribution with shape $k/2$ and rate $1/2$. If $Q \sim \chi^2_k$, then $E[Q] = k$ and $\operatorname{Var}(Q) = 2k$. If $Q_1 \sim \chi^2_{k_1}$ and $Q_2 \sim \chi^2_{k_2}$ are independent, then $Q_1 + Q_2 \sim \chi^2_{k_1 + k_2}$. For an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample of size $n$ with sample variance $S_n^2$, $(n - 1) S_n^2 / \sigma^2 \sim \chi^2_{n - 1}$.

Explanations

L1 human

Square $k$ independent standard normal numbers and add them: the result has the chi-squared distribution with $k$ degrees of freedom. Its mean is $k$, and it has values in $[0, \infty)$ only.