Moment generating functions evaluated at points other than the origin

Question

If we have a moment generating function $M_X(s)$, we know we can take the derivative twice and evaluate it at $s=0$ to find the second moment of the associated PDF. However, is there any pattern or behavior we can expect if we evaluate the second derivative of $M_X(s)$ at a point other than $0$. Basically I'm asking if there should be some predictable way the second derivative of $M_X(s)$ behaves at points centered around a point other than the origin.

Answer 1

Well, just compute it: $$ \left.\frac{d^2}{ds^2}\right|_{s=s_0}\mathbb E(e^{sX})=\mathbb E(X^2 e^{s_0X}),$$ (provided that one can exchange derivatives and integrals, etc, etc...). Of course this is most useful when $s_0=0$ (as is often the case with formulas involving the exponential function), because in that case we have in the right hand side the second moment of $X$.