## Encyclopedia > Moment-generating function

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# 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:
$E\left(X^n\right)=M_X^{(n)}(0)=\left.\frac{d^n}{dt^n}\right|_{t=0} M_X(t).$
Related concepts include the characteristic function and the probability-generating function.

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