We have a discrete random variable $ X $ with distribution function $ F_X $ and $ mathrm {range} (X) = {1, 2, dots, U } subset mathbb {N} $ (that is, strictly positive integers with an upper bound $ U $).

We define $ X _ { geq p} $ as $ X $ The prerequisite is that you are on top $ Q (p) $, from where $ Q $ is the quantum function. This is, $ X _ { geq p} Excess { smaller Text {def}} {=} X center X in links {x: p leq F_ {X} (x) right } $,

I try to find a narrow upper limit $ mathbb {E} (X _ { geq p} / X) $. $ frac {1} {2} <p <1 $,

I started a little naïve and managed to get tied up $ mathbb {E} X _ { geq p} / mathbb {E} X $ only to know, I can not justify $ mathbb {E} (X _ { geq p} / X) leq mathbb {E} X _ { geq p} / mathbb {E} X $ although I think it makes sense in this case.

Can someone point me in the right direction? Thank you very much!

**additional note:** In other words. Suppose I guessed an upper limit $ A $ and a parameter $ 0 leq delta leq 1 $, How can I *check* if the following applies (I can try it out $ X $ as much as necessary):

$ P ( mathbb {E} (X_ {≥p} / X)> = A) < delta $?

**Note**: I also accidentally posted this in https://math.stackexchange.com/questions/3329721/expectation-of-upper-quantile-proportion and then marked that it should be moved here by an administrator.