If a r.v. $X < a$, does it imply $EX < a$?
- If not, why is it different from what I know: If a r.v. $X \leq a$, it implies $EX \leq a$, proved by replacing $X$ with $a$ as the integrand.
- Note that $a \in \mathbb R$. If it allows that $a= \infty$, the answer is no: if $X$ has a Pareto distribution with $α=1$, then $X < \infty$, but $EX = \infty$, from http://en.wikipedia.org/wiki/Pareto_distribution and https://stats.stackexchange.com/a/91515/1005