# Probability – upper limit of the expectation of the upper quantile ratio

We have a collection $$boldsymbol {S}$$ from $$n$$ discrete random variables $$X_1$$. $$X_2$$. $$dots$$. $$X_n$$ $$overset { small text {i.d.}} { small sim}$$ $$mathcal {D}$$, from where $$mathcal {D}$$ is a distribution over $${0, 1, ldots, U } subset mathbb {N}$$ with cumulative distribution function $$F_ mathcal {D}$$,

We define the sub listing that contains only the values ​​in $$boldsymbol {S}$$ that's up $$Q (p)$$, from where $$Q$$ is the quantum function. This is:

$$boldsymbol {S} _ { geq p} overset { smaller text {def}} {=} left {X: X in boldsymbol {S} text {and} p leq F _ { mathcal {D}} (X) right }$$

(in words: $$X in boldsymbol {S} _ { geq p}$$ if and only if it is $$p$$ Population less or equal)

(below we mark $$pmb { sum} boldsymbol {C}$$ as the sum of all items in the collection $$boldsymbol {C}$$)

Is the following true?

$$forall n, ; mathbb {E} left ( frac { pmb { sum} boldsymbol {S} _ { geq p}} { pmb { sum} boldsymbol {S}} right) le frac { mathbb {E} pmb { sum} boldsymbol {S} _ { geq p}} { mathbb {E} pmb { sum} boldsymbol {S}}$$

Note This is an extracted step of another question Expecting the upper quantile part.
This question is simpler and more targeted. (+ if the above statement is true, the previous question is solved)