11.3 The $F$ Distribution

Parameters : $𝜽=({\nu }_1,{\nu }_2)$ with ${\nu }_1>0$ and ${\nu }_2>0$ called the degrees of freedom.

1. $f_X(x;𝜽)=\frac{{\nu }_1^{\frac{{\nu }_1}{2}}{\nu }_2^{\frac{{\nu }_2}{2}}{x}^{\frac{{\nu }_1}{2}-1}}{B(\frac{{\nu }_1}{2},\frac{{\nu }_2}{2}){({\nu }_{1}x+{\nu }_2)}^{\frac{{\nu }_1+{\nu }_2}{2}}}$ for ,

2. $𝖤\left[X\right]=\frac{{\nu }_2}{{\nu }_2-2}$ when ${\nu }_2>2$,

3. $\mathrm{𝖵𝖺𝗋}\left[X\right]=\frac{2{\nu }_2^2({\nu }_1+{\nu }_2-2)}{{\nu }_1{({\nu }_2-2)}^2({\nu }_2-4)}$ when ${\nu }_2>4$.

We write $X\sim F_{{\nu }_1,{\nu }_2}$.

Transformations: If $V_1$ and $V_2$ are independent with $V_1$ ${\chi }_{{\nu }_1}^2$-distributed and $V_2$ ${\chi }_{{\nu }_2}^2$-distributed, then

is $F_{{\nu }_1,{\nu }_2}$-distributed.

Usage: Used in statistics as the distribution of the test statistic in analysis of variance (ANOVA).

Other: Named after the eminent English statistician R.A. Fisher.