moment-generating function

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The moment-generating function of a random variable X involves a variable, usually denoted by t, and is defined to be the expectation of etX. It is usually denoted by MX(t), or M(X, t). Expanding MX(t) in powers of t gives where μr′ is the rth moment of X about the origin. By virtue of the Maclaurin series, μr′ is the value, when t=0, of the rth derivative with respect to t of MX(t). See also probability-generating function.

Subjects: Probability and Statistics.

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