Identity of moment-generating functions Are there any non-identical distributions which happen to have the same moment-generating function?
 A: Yes.
In an exercise, Stuart & Ord (Kendall's Advanced Theory of Statistics, 5th Ed., Ex. 3.12) quote a 1918 result of TJ Stieltjes (which apparently appears in his Oeuvres Completes,):

If $f$ is an odd function of period $\frac{1}{2}$, show that
$$\int_0^\infty x^r x^{-\log x} f(\log x) dx = 0$$
for all integral values of $r$.  Hence show that the distributions
$$dF = x^{-\log x}(1 - \lambda \sin(4\pi \log x))\ dx, \quad 0 \le x \lt \infty;\quad 0 \le |\lambda| \le 1,$$
have the same moments whatever the value of $\lambda$.

(In the original, $|\lambda|$ appears only as $\lambda$; the restriction on the size of $\lambda$ arises from the requirement to keep all values of the density function $dF$ non-negative.)  The exercise is easy to solve via the substitution $x = \exp(y)$ and completing the square.  The case $\lambda=0$ is the well-known lognormal distribution.

The blue curve corresponds to $\lambda=0$, a lognormal distribution.  For the red curve, $\lambda = -1/4$ and for the gold curve, $\lambda = 1/2$.
