$\chi^2$ Distribution

• If $Z$ is a standard normal r.v., the distribution of $U = Z^2$ is called the chi-square distribution with 1 degree of freedom.
• If $U_1,U_2...$ are independent chi-square r.v. with 1 degree of freedom, the distribution of $V = U_1+U_2+...$ is called the chi-square distribution with n degrees of freedom and is denoted by $\chi_n^2$
• $\frac{(n-1)S^2}{\sigma^2}$ has a $\chi^2(n-1)$ distribution

$t$ distribution

If $Z \sim N(0, 1)$ and $U \sim \chi^2_n$ and $U$ are independent, then the distribution of $\frac{Z}{\sqrt{U/n}}$ is called the $t$ distribution with n degrees of freedom.

• if $X_1,...,X_n$ are a random sample from a $N(\mu,\sigma^2)$, we know that $\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)$
• but if $\sigma$ is unknown, we can use $S$ to replace $\sigma$, and $\frac{\bar{X} - \mu}{S_n/\sqrt{n}} \sim t(n-1)$

t-distribution converges to a normal distribution if the freedom degree approaches to infinity.

Property

• $t(1) \sim Cauchy$
• $E(t(p)) = 0$
• $Var(T(p)) = \frac{p}{p-2}$

$F$ distribution

Let $U$ and $V$ be independent $\chi^2$ r.v. with $m$ and $n$ degrees of freedom, respectively. The distribution of

$W = \frac{U/m}{V/n}$

is call the $F$ distribution with $m$ and $n$ degrees of freedom and is denoted by $F_{m,n}$

• $U = \frac{(n-1)S_x^2}{\sigma ^2_x} \sim \chi^2(n-1)$
• $V = \frac{(n-1)S_y^2}{\sigma ^2_y} \sim \chi^2(m-1)$
• $\frac{U/n-1}{V/m-1} = \frac{S_x^2 / \sigma ^2_x}{S_y^2 / \sigma ^2_y} \sim F(m,n)$

Property

• A variance ratio may have an F distribution even if the parent populations are not normal. It is enough that their pdf are symmetry functions.

• if $X \sim F_{p,q} ,$then $\frac{1}{X} \sim F_{q,p}$

• if $X \sim t(q)$, then $X^2 \sim F_{1,q}$