If X is a nonnegative random variable with mean 25, what can be said about:
$E[X^3]$
$E[\sqrt{X}]$
$E[logX]$
$E[e^{-X}]$
My guess is that we need to know more information, like variance, to solve these expectations, but I have a feeling there might be some trick to them.
You're definitely right that we need more information. The distribution of $X$ would be really helpful.
1.) $\mathbb{E}(X^3)$ isn't necessarily guaranteed to exist if we only know the mean exists. See the Cauchy Distribution
2.) $\mathbb{E}(\sqrt{X})$ is only defined if $X$ is nonnegative almost surely. If it is, then by Jensen's Inequality we can say $\mathbb{E}(\sqrt{X}) \leq \sqrt{25}= 5$
3.) Again, all we can really say about this without any info is what's given by Jensen's inequality: $\mathbb{E}(\log X) \leq \log \mathbb{E}(X) = \log(25)$
4.) We can apply Jensen's inequality to this as well, but its also the Moment Generating Function of $X$ evaluated at $s=-1$. Not all random variables have defined moment generating functions, however. See log-normal distribution.