Monotone tail and moment ratio properties of Student’s family of distributions

Let G p denote the tail function of Student’s distribution with p degrees of freedom. It is shown that the ratio G q ( x )/ G p ( x ) is decreasing in x > 0 for any p and q such that 0 < p < q ≤ ∞ . Therefore, G q ( x ) < G p ( x ) for all such p and q and all x > 0. Corollaries on the monotonicity of (generalized) moments and ratios thereof are also given.