Suppose that, for a discrete, simple, and positive random variable $X$, it happens with probability $ 1 - 1/n$ that $X\le f(n)$, for some function $f(n) \in o(n)$. I don't know anything else about the actual distribution of $X$.
Is it possible to upper bound $E[X]$ in terms of $f(n)$? The intuition I have is that if $X$ is less than $f(n)$ with high probability, then clearly it's expectation should also be less than $f(n)$.
This seems to be like a converse statement of Markov's inequality where we know something about the expectation and use it to bound the random variable itself.