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In [[probability theory]] and [[statistics]], the '''factorial moment generating function''' (FMGF) of the [[probability distribution]] of a [[real number|real-valued]] [[random variable]] ''X'' is defined as
:<math>M_X(t)=\operatorname{E}\bigl[t^{X}\bigr]</math>
for all [[complex number]]s ''t'' for which this [[expected value]] exists. This is the case at least for all ''t'' on the [[unit circle]] <math>|t|=1</math>, see [[characteristic function (probability theory)|characteristic function]]. If ''X'' is a discrete random variable taking values only in the set {0,1, ...} of non-negative [[integer]]s, then <math>M_X</math> is also called [[probability-generating function]] (PGF) of ''X'' and <math>M_X(t)</math> is well-defined at least for all ''t'' on the [[closed set|closed]] [[unit disk]] <math>|t|\le1</math>.
The factorial moment generating function generates the [[factorial moment]]s of the [[probability distribution]].
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