Moment-generating function

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

$M_{X}(t)=E\left(e^{tX}\right).$

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

$E\left(X^{n}\right)=M_{X}^{(n)}(0)=\left.{\frac {d^{n}}{dt^{n}}}\right|_{t}=0M_{X}(t).$