Finding E(x) from E(ln(x)).

Question

Say you have $\operatorname{E}[\ln(x)]=\mu$, is there a way to find $\operatorname{E}[x]$?

This seems like a really simple question but I can't figure it out. Any help would be appreciated.

Answer 1

No--there are infinitely many possible values for $\mathbb E[X]$ for a given value of $\mathbb E[\ln X]$. For example, if $X$ has a Lognormal distribution with parameters $\mu, \sigma^2$, $\mathbb E[\ln X] = \mu$, but $\mathbb E[X]$ also depends on $\sigma^2$, and can be arbitrarily large.

You can bound $\mathbb E[X]$ below by $\exp\{\mathbb E [\ln X]\}$ by Jensen's inequality, but that's the best you can do in general.