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moment generating function (Definition)

Given a random variable $ X$, the moment generating function of $ X$ is the following function:

$ M_X(t) = E[e^{tX}]$ for $ t \in R$ (if the expectation converges).

It can be shown that if the moment generating function of $ X$ is defined on an interval around the origin, then

$ E[X^k] = M_X^{(k)}(t) \vert _{t=0} $

In other words, the $ k$th-derivative of the moment generating function evaluated at zero is the $ k$th moment of $ X$.



"moment generating function" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: characteristic function, cumulant generating function


Attachments:
moment generating function of the sum of independent random variables (Corollary) by me_and
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Cross-references: moment, origin, interval, converges, expectation, function, random variable
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This is version 5 of moment generating function, born on 2001-10-26, modified 2006-09-18.
Object id is 512, canonical name is MomentGeneratingFunction.
Accessed 22658 times total.

Classification:
AMS MSC60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)

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