Parametric distribution where the mean of a constant to the power of the random variable is an easy to use function Let $X$ be a random variable with distribution $F\left(x;\theta\right)$ where $\theta$ are the parameters of the distribution. Let $c$ be a constant.
Define:
$h\left(c,\theta\right) = \mathbb{E}\left[ c^{X}\right]$.
Are there are any distributions of $X$ for which $h\left(c,\theta\right)$ is a known, easy-to-evaluate function?
 A: Just about anything with an explicit simple-form mgf should work.
Note that $E(c^X)=E(e^{kX})$ for $k=\ln c$.
So up to substitution of symbols, this is an mgf -- pretty well-laid-out territory.
The most obvious instance will probably be the lognormal distribution (where $X$ is normal, so $kX$ is normal, so $c^X=e^{kX}$ is lognormal, and its expectation is well known, or readily derived).
However, lots of other candidates will work just fine since lots of mgfs are readily available (e.g. via wikipedia). You can even find lists of mgfs in many books or online.
There's a short list at the wikipedia page on the moment generating function.
So for example, for the continuous uniform on $(0,b)$ (taking $a=0$ from the listed $U(a,b)$ item at that link), we would have the mgf 
$$E(e^{tX})=\frac{e^{tb} - 1}{tb}$$
consequently, 
$$E(c^X) = E(e^{kX}) = \frac{e^{kb} - 1}{kb}=\frac{e^{b\ln c} - 1}{b\ln c}\,.$$
Similar substitutions work for other cases. Many individual distribution pages on Wikipedia list an mgf.
Of course, it's often worth learning how to find those mgfs in any case, but finding some distributions for which you can evaluate $E(c^X)$ is very easy.
