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

Understand

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\le\infty.
• 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.