Linearity of maximum function in expectation I was solving an exercise for a probability theory course and stumbled upon the following problem.
Given a continuous random variable $X$, and $\max(a,b) = a$ if $a > b$ and $b$ otherwise, is
$$
E[\max(X, 0)] = \max(E[X], 0)
$$
true in general?

No, it's not. If $E[X] > 0$ then
$$
max(E[X], 0) = E[X]
$$
so you would need
$$
E[X] = E[max(X, 0)]
$$
That can easily be disproved with simple counterexamples.
If $E[X]\le0$ then
$$
max(E[X], 0) = 0
$$
so you would need to have
$$
E[max(X, 0)]=0
$$
That would hold only for constant $X$ equal to $0$.
TL;DR Both ways, it can be easily disproved.
