Probstat/notes/chi-squared distribution

This is part of probstat.

Definition

Let $Z_{1},Z_{2},\ldots ,Z_{n}$ be independent unit normal random variables. A random variable

$X=Z_{1}^{2}+Z_{2}^{2}+\cdots +Z_{n}^{2},$

is called a chi-squared random variable with $n$ degree of freedom. We also write

$X\sim \chi _{n}^{2}.$

Wikipedia has a nice article on chi-squared distribution which also includes plots of its pdf and cdf.

Properties

Here we states important properties of the chi-squared distribution without proofs.

Expectations and variances

If $X$ is chi-squared with $n$ degree of freedom, we have that its expectation

$E[X]=n,$

and its variance

$Var(X)=2n.$

Sample variances

The chi-squared distribution is very important when we want to reason about the sample variances (See notes). Under that settings, we sample $X_{1},X_{2},\ldots ,X_{n}$ from a normal population whose mean is $\mu$ and variance is $\sigma ^{2}$.

Recall that ${\frac {X_{i}-\mu }{\sigma }}$ is unit normal. Therefore,

$\sum _{i=1}^{n}\left({\frac {X_{i}-\mu }{\sigma }}\right)^{2}={\frac {1}{\sigma ^{2}}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}\sim \chi _{n}^{2}.$

Links

• Wikipedia article on the chi-squared distribution