Definition of Moment-Generating Function: The moment-generating function is denoted M(t) or M_X(t), and describes a probability distribution. A moment-generating function is defined for any random variable X with a pdf f(x).

M(t) is defined to be E[e^tX], which is the integral from minus infinity to infinity of e^tXf(x).

A use for these is that the t^th moment of X is M^(t)(0), that is the t^th derivative of M() at zero. (Econterms)

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