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In probability theory, the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments are useful for studying non-negative integer-valued random variables.〔D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. I''. Probability and its Applications (New York). Springer, New York, second edition, 2003.〕 and arise in the use of probability-generating functions to derive the moments of discrete random variables. Factorial moments serve as analytic tools in the mathematical field of combinatorics, which is the study of discrete mathematical structures. ==Definition== For a natural number , the -th factorial moment of a probability distribution on the real or complex numbers, or, in other words, a random variable with that probability distribution, is : where the is the expectation (operator) and : is the falling factorial, which gives rise to the name, although the notation varies depending on the mathematical field. Of course, the definition requires that the expectation is meaningful, which is the case if or . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「factorial moment」の詳細全文を読む スポンサード リンク
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