Moment-generating function

In probability theory and statistics, the moment-generating function of a random variable X is

<math>M_X(t)=E\left(e^{tX}\right).</math>

The moment-generating function generates the moments of the probability distribution, as follows:

<math>E\left(X^n\right)=M_X^{(n)}(0)=\left.\frac{d^n}{dt^n}\right|_{t=0} M_X(t).</math>

Related concepts include the characteristic function and the probability-generating function.