Continuous random variable

Chi-squared distribution

A chi-squared distributed random variable is the sum of squares of independent and identically distributed with standard normal distributions. prob

This turns out to be a special case of the Gamma distribution, namely .

Proof

Let for , i.e. . Then

thus

so . Thus by ^Q1, the claim is proven.

Properties

Additional properties

1. Let be a random sample of variable independently distributed according to the normal distribution . Then the sample variance is distributed such that

\begin{align*} \frac{(n-1)S_{n}^2}{\sigma^2} \sim \chi^2_{n-1} \end{align*}