Given a random variable $X$ , the moment generating function of $X$ is the following function:

$M_X(t) = E[e^{tX}]$ for $t \in R$ (if the expectation converges).

It can be shown that if the moment generating function of $X$ is defined on an interval around the origin, then

$E[X^k] = M_X^{(k)}(t) |_{t=0} $

In other words, the $k$ th-derivative of the moment generating function evaluated at zero is the $k$ th moment of $X$ .