# Probability – expectation of the upper quantile component

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} ,

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.