Comparisons of Conditional Expectations

Question

Suppose $X_1,X_2,...,X_m$ are $m$ independent random variables. Let $L(k)$ be the $k$ largest random variables among them. Do we always have $$E[X_i\mid X_i \in L(k)]\ge E[X_i\mid X_i \not \in L(k)]?$$

Answer 1

Let $Y$ denote the $k^\text{th}$ largest value among all of the variables except for $X_i$. Then $E[X_i\mid X_i\in L(k)]=E[X_i\mid X_i\ge Y]$, where $X_i$ and $Y$ are independent. Therefore, your question is equivalent to:

If $X$ and $Y$ are independent, is $E[X\mid X\ge Y]\ge E[X\mid X<Y]$?

Obviously, this is true when $Y$ is deterministically equal to the real number $y$, since $E[X\mid X\ge y]\ge y\ge E[X\mid X<y]$. Therefore, it is true when you average over $y$ according to the distribution of $Y$.

To add some more detail, letting $f(y)$ be the pdf of $Y$, then $$ E[X\mid X\ge Y]=\int E[X\mid X\ge y]f(y)\,dy\ge \int E[X\mid X<y]f(y)\,dy = E[X\mid X<Y] $$