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In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have identical cumulants as well, and similarly the cumulants determine the moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. Just as for moments, where ''joint moments'' are used for collections of random variables, it is possible to define ''joint cumulants''. ==Definition== The cumulants of a random variable are defined via the cumulant-generating function , which is the natural logarithm of the moment-generating function: : The cumulants are obtained from a power series expansion of the cumulant generating function: : This expansion is a Maclaurin series, so that th cumulant can be obtained by differentiating the above expansion times and evaluating the result at zero:〔Weisstein, Eric W. "Cumulant". From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/Cumulant.html〕 : If the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later. 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「cumulant」の詳細全文を読む スポンサード リンク
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