I have a positive random variable $X$, can I prove any relation between $\exp(\mathbb{E}[X])$ and $\mathbb{E}[\exp(X)]$. To elaborate, I don't have the PDF for $X$ but I do have its mean and from it I want to get $\mathbb{E}[\exp(X)]$.
So, Can I prove any relation between both quantities? I believe there is no relation, but can I use $\exp(\mathbb{E}[X])$ to obtain a bound on $\mathbb{E}[\exp(X)]$? and what would be the conditions for such a bound to hold? also, how tight is this bound?
See Jensen's inequality. For a convex function $\exp$, we see that this implies $$\exp \operatorname{E}[X] \le \operatorname{E}[\exp X].$$ Conditions on when equality is attained are left as an exercise for the reader.