F-distribution

Definition

The $F$-distribution with $n$ and $m$ degrees of freedom is the distribution of

$$Z=\frac{X/n}{Y/m}$$

where $X\sim {\chi }_n^2,Y\sim {\chi }_m^2$ (see Chi-Squared Distribution), we denote this as

$$Z\sim F_{n,m}$$

Mean and Variance

• Mean: $\frac{m}{m-2}$ for $m>2$
• Variance: $\frac{2m^2(n+m-2)}{n(m-2{)}^2(m-4)}$ for $m>4$

Properties

If $Z\sim F_{n,m}$, then $\frac{1}{Z}\sim F_{m,n}$

If $T\sim t_n$ (Student's t-distribution), then $T^2\sim F_{1,n}$

As $m\to \infty$, $n{F}_{n,m}\stackrel{d}{\to }{\chi }_n^2$