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Given a random variable  , the moment generating function of  is the following function:
![$ M_X(t) = E[e^{tX}]$](http://images.planetmath.org:8080/cache/objects/512/l2h/img3.png "$ M_X(t) = E[e^{tX}]$") for  (if the expectation converges).
It can be shown that if the moment generating function of  is defined on an interval around the origin, then
![$ E[X^k] = M_X^{(k)}(t) \vert _{t=0} $](http://images.planetmath.org:8080/cache/objects/512/l2h/img6.png "$ E[X^k] = M_X^{(k)}(t) \vert _{t=0} $")
In other words, the  th-derivative of the moment generating function evaluated at zero is the  th moment of  .
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