Deducing a probability distribution from its moment-generating function It's pretty trivial to get a moment-generating function from a p.d.f. (provided $\sum e^{tx}f(x)$ isn't too difficult to evaluate), but since moment-generating functions uniquely determine a probability distribution function, is there a way to "back-generate" the p.d.f from the m.g.f.?
Edit: I'm talking about a discrete distribution.

The question is to inverse a Laplace/Fourier transform. I take the example of a discrete distribution $f(n)$ on the natural numbers with moment-generating-function
$$
M(t) = \sum_{n=0}^\infty f(n)e^{nt}
$$
with radius of convergence $R \geq 1$. Fourier inversion here is
$$
f(n) = \frac{1}{2\pi}\int_0^{2\pi}M(i\theta)e^{-in\theta}\,d\theta
$$
If you prefer to stay in the real realm, there is an interesting formula due to Post.
A related formula would be:
$$
f(n) = \left.\frac{d^n}{dt^n}M(\log t)\right|_{t=0}
$$
