Moment-generating function |
In probability theory and statistics, the moment-generating function of a random variable X is
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The moment-generating function generates the moments
of the probability distribution, as follows:
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If X has a continous probability
density function f(x) then the moment generating function is given by
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where mi is the ith moment.
Regardless of whether probability distribution
is continuous or not, the moment-generating function is given by the Riemann-Stieltjes integral
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where F is the cumulative
distribution function.
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.
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