Inspired by Xian Ng

I assume "False for most X" means "False for some, but not all, X".

Just observe that it doesn't work for the random variable $X : \{1, 2\} \to \left\{ \frac{1}{2} \right\}$ ; that is, a random variable giving $1$ and $2$ uniformly randomly. (We have $E[X] = \frac{3}{2}$ , so $\frac{1}{E[X]} = \frac{2}{3}$ , but $E \left[ \frac{1}{X} \right] = \frac{3}{4}$ .)

On the other hand, it does work for the trivial random variable $X : \{1\} \to \{1\}$ , since $E[X] = E \left[ \frac{1}{X} \right] = 1$ .

So it doesn't work for some $X$ , but works for some others.

This was inspired by me??? How so? I didn't even understand the question looool Oh and since this is supposed to be a solution I intuitively deduced that a straightforward answer probably is not correct so i guess the "most" ans lel