In probability theory
, the moment-generating function
of a random variable X
wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.
If X has a continuous probability density function f(x) then the moment generating function is given by
where is the ith moment. is just the two-sided Laplace transform of f(x).
Regardless of whether the 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, ..., Xn is a sequence of independent (and not necessarily identically distributed) random variables, and
where the ai are constants, then the probability density function for Sn is the convolution of the probability density functions of each of the Xi and the moment-generating function for Sn is given by
For vector-valued random variables X with real components, the moment-generating function is given by
where t is a vector and is the dot product.
Provided the moment-generating function exists in an open interval around t = 0, the nth moment is given by
If the moment generating function is finite in such an interval, then it uniquely determines a probability distribution.
Related to the moment-generating function are a number of other transforms that are common in probability theory, including the characteristic function and the probability-generating function.
The cumulant-generating function is the logarithm of the moment-generating function.