The Chi-squared, t, and F distributions

 
[1] The Chi-squared ($\chi^2$) distribution
Definition
If $X_{1},...,X_{p}$ are iid N(0,1) and $X=(X_{1},...,X_{p})^T \sim N_{p}$(0,I), (I is the pxp identity matrix)
$V=\left \| X \right \|^2=X^TX$ (The squared length of X), then $V = X_{1}^2+...+X_{p}^2 \sim \chi_{p}^2$
 

[2] The t-distribution
Definition 
Let X~ N(0,1) and $V \sim \chi_{n}^2$  be independent random variables, 
then $T= \frac{Z}{\sqrt{V/n}} \sim t_{n}$ with n degrees of freedom.  


[3] The F distribution
Definition 
Let $U \sim \chi_{m}^2$  and $V \sim \chi_{n}^2$ be independent random variables,
the variables $W=\frac{U/m}{V/n} \sim F_{m,n}$ , the ratio of independent chi-squared variables.