Task

The $F$-distribution is also called the variance ratio distribution. It is generally used for the analysis of variance.

Let $U\sim\chi^2_{v_1}$ and $V\sim\chi^2_{v_2}$, if they’re independent, then $F=\frac{U/ v_1}{V / v_2}$ has a $F$-Distribution.

• Given $S^2_1$ and $S^2_2$ are variances of independent random samples with size $n_1$ and $n_2$ taken from normal populations with variances $\sigma^2_1$ and $\sigma^2_2$ respectively, then $F = \frac{S^2_1 / \sigma^2_1}{S^2_2 / \sigma^2_2}$ has a $F$-distribution with $v_1=n_1-1$ and $v_2=n_2-1$ DOF.

Writing $f_a(v_1,v_2)$ for $f_a$ with $v_1$ and $v_2$ degrees of freedom, we have

$$ f_{1-a}(v_1,v_2) = \frac{1}{f_a(v_2,v_1)} $$

e.g. $f_{0.95}(6,10)=\frac{1}{f_{0.05}(10,6)}$

• $F$-Distribution Table