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

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

If X has a continuous probability density function f(x) then the moment generating function is given by

where m_i is the ith moment.

Regardless of whether probability distribution is continuous or not, the moment-generating function is given by the Riemann-Stieltjes integral

where F is the cumulative distribution function.

If X₁, X₂, ..., X_n is a sequence of independent (and not necessarily identically distributed) random variables, and

where the a _i are constants, then the probability density function for S _n is the convolution of the probability density functions of each of the X _i and the moment-generating function for S _n is given by

Related concepts include the characteristic function, the probability-generating function, and the cumulant-generating function. The cumulant-generating function is the logarithm of the moment-generating function.