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)