Crossing properties of F distributions

The crossing properties of F distributions are discussed. The results show that the F distribution stochastically decreases in its denominator degrees of freedom if the numerator degrees of freedom m is either one or two, and the difference of two F distribution functions with the same numerator degrees of freedom m [greater-or-equal, slanted] 3 has number of sign changes of one. It is also shown that the upper percentile of Student t distribution decreases in its degrees of freedom.