Suppose the random variable $W$ has an $F$-distribution $F(r_1,r_2)$. My probability and statistics textbook writes: if $\alpha = P[W \geq F_{\alpha}(r_1,r_2)]$,
then $\alpha = P[W \leq F_{1-\alpha}(r_1,r_2)] = P[\frac {1}{W} \geq \frac {1} {F_{1-\alpha}(r_1,r_2)}]$ (not sure why this last equality is true)
and moreover that $P[\frac {1} {w} \leq F_{\alpha}(r_2,r_1)] = \alpha$. (why does reversing the order of $(r_1,r_2)$ have this effect?)
Any insights into how these equalities are derived much appreciated.
Note that one of the definitions of $F(n_1, n_2)$ distribution is by independent two scaled $\chi^2$ random variable. Namely, if $ U_1 \sim \chi^2(n_1)$ and $U_2 \sim \chi^2(n_2)$, thus $$ F(n_1, n_2) = \frac{U_1/n_1}{ U_2/n_2}, $$ and $$ F(n_2, n_1) = \frac{U_2/n_2}{ U_1/n_1} =\frac{1}{F(n_1, n_2)}\, . $$