# Chi-square distribution

A sum of the squares of $k$ independent standard normal random variables

$k$
$$Z_1^2 + Z_2^2 + \cdots + Z_k^2 \;\sim\; \chi^2(k)$$
$$f(x) = \begin{cases} \frac{x^{\frac{k}{2}-1}e^{-\frac{x}{2}}}{2^{\frac{k}{2}}\Gamma\big(\frac{k}{2}\big)}, & \text{if x > 0,} \\ 0, & \text{otherwise.} \end{cases} \\[24pt] \text{where k is the degree of freedom (the number of the squares)} \\ \\ \text{and \Gamma\bigg(\frac{k}{2}\bigg) is the gamma function.}$$