Expectation of maximum of two random variables inequality

Question

I have a question about the expectation of maximum of two variables.

Let $X_1$, $X_2$, and $Y$ be random variables.

Suppose $E[X_1] > E[X_2]$. Then is it necessarily the case that $E[\max(X_1, Y)] > E[\max(X_2, Y)]$?

If there is any assumption that has to be made, please let me know.

Answer 1

No. Here's a counterexample. Consider the following three independent random variables:

• let $X_1$ be a constant-valued random variable, which is always 1.
• let $X_2$ be 3 with probability $1/4$ and 0 otherwise.
• let $Y$ be 1 with probability $1/2$ and 0 otherwise.

Then $E(X_1) = 1, E(X_2) = 3/4$.

But since $X_1 \ge Y$, we have $\max(X_1, Y) = X_1$ and so $E(\max(X_1, Y)) = 1$. But $\max(X_2, Y)$ is 3 with probability $1/4$, 1 with probability $3/8$ and 0 otherwise, so $E(\max(X_2, Y))= 9/8$.