Moment-generating function

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

wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.

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 interval around t = 0, the nth moment is given by

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

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.

See also

de:Momenterzeugende Funktion fa:تابع مولد ممان ko:모멘트생성함수 nl:Momentgenererende functie su:Fungsi nu ngahasilkeun momen he:פונקציה יוצרת מומנטים