# Chi-Squared Distribution $χ_{2}$

Very few real-world observations follow a chi-square distribution. The main purpose of chi-square distributions is Hypothesis Testing, not describing real-world distributions.

Definition 1:

Let $n$ be a positive integer. Then we say $W∼χ_{2}(n)$ if

$W=Z_{1}+Z_{2}+⋯+Z_{n}$
where

- $Z_{i}∼N(0,1)$, $Z_{i}$’s are independent
- $n$ a parameter that represents the Degrees of Freedom

Definition 2 using the PDF: $f(w)=2_{2n}Γ(2n )1 w_{2n−1}e_{2−w},w≥0$ where

- $Γ(α)$ is the Gamma Function

Expectation and Variance

- $E(X)=n=df$ where $df$ is the Degrees of Freedom
- $Var(X)=2n=2df$

If we apply CLT, we get that as $n→∞$, $W∼N(n,2n)$

- $n=1⟹Z_{2}$
- $n=2⟹Exp$
- $⋯⟹χ_{2}$ table
- $n=∞⟹Normal$

### Chi-Squared Table

### Examples

So in here, unlike in the Z-Table, you are given a degree of freedom and a probability, and this table gives you the $X$ value.

Since $χ_{2}$ is one-tailed, then you just check the 95%,

Ex1: df =15, percentile = 0.9, you find that x = 22.307 Ex2: Let $W∼χ_{2}(11)$, Find $a$ and $b$ such that $P(a≤w≤b)=0.95$

You can consider the probabilities $0.025$ and $0.975$, so you have $P(3.816≤w≤21.92)=0.95$.