Chi-Squared Distribution

if the random variable $Z$ has a standard normal distribution, $Z^{2}$ has a $χ^{2}$ distribution with one degree of freedom
- if $Z_{1}, Z_{2}, \dots, Z_{ν}$ are independent normal random variables then $Z_{1}^{2}, Z_{2}^{2}, \dots, Z_{ν}^{2}$ has a $χ^{2}$ distribution with $ν$ degrees of freedom
  - $μ = ν, σ^{2} = 2 ν$
  - mode occurs at $χ = d f - 2$ for distributions with at least 2 dfs
    - mode = highest occurence
the test statistic in a $χ 2$ test will have an approximate $χ 2$ distribution
as the degrees of freedom $\to \infty$ the $χ^{2}$ approximates a normal distribution with mean $= ν$

PDF

A random variable $X$ has a chi-square distribution with $v$ degrees of freedom, $X \sim χ_{v}^{2}$ if and only if it has the pdf

f (x) = \frac{1}{2^{v / 2} Γ (v / 2)} x^{v / 2 - 1} e^{- x / 2} for x \geq 0

χ_{v}^{2} \equiv G a m m a (\frac{v}{2}, 2)

Mean

E (X) = (\frac{v}{2}) (2) = v

Variance

V a r (X) = (\frac{v}{2}) (2^{2}) = 2 v

If a random variable X has a standard normal distribution (with mean
0 and unit variance), then a random variable Y = X2 has a χ2 distri-
bution with 1 df (denoted by χ2
1). If Y is the sum of k independent
standard normal squares, Y has a χ2 distribution with k df ( χ2
k).
In other words, if Y is the sum of k independent χ2
1’s, Y has a χ2
distribution with k df ( χ2
k).

If $x_{1}, x_{2}, \dots, x_{k}$ are independent random variables, $X_{i} \sim N (0, 1)$

Y = X_{1}^{2} + X_{2}^{2} + \dots + X_{k}^{2} \sim χ_{k}^{2}

$v = k =$ number o square of normal sum