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In mathematics, a moment is a specific quantitative measure, used in both mechanics and statistics, of the shape of a set of points. If the points represent mass, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the points represent probability density, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second moment is the variance, the third moment is the skewness, and the fourth moment (with normalization and shift) is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. For a bounded distribution of mass or probability, the collection of all the moments (of all orders, from to ) uniquely determines the distribution. ==Significance of the moments== The -th moment of a real-valued continuous function ''f''(''x'') of a real variable about a value ''c'' is : It is possible to define moments for random variables in a more general fashion than moments for real values—see moments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with ''c'' = 0. For the second and higher moments, the central moments (moments about the mean, with ''c'' being the mean) are usually used rather than the moments about zero, because they provide clearer information about the distribution's shape. Other moments may also be defined. For example, the -th inverse moment about zero is and the -th logarithmic moment about zero is The -th moment about zero of a probability density function ''f''(''x'') is the expected value of and is called a ''raw moment'' or ''crude moment''.〔http://mathworld.wolfram.com/RawMoment.html Raw Moments at Math-world〕 The moments about its mean are called ''central'' moments; these describe the shape of the function, independently of translation. If ''f'' is a probability density function, then the value of the integral above is called the -th moment of the probability distribution. More generally, if ''F'' is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the -th moment of the probability distribution is given by the Riemann–Stieltjes integral : where ''X'' is a random variable that has this cumulative distribution ''F'', and is the expectation operator or mean. When : then the moment is said not to exist. If the -th moment about any point exists, so does the -th moment (and thus, all lower-order moments) about every point. The zeroth moment of any probability density function is 1, since the area under any probability density function must be equal to one. 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「Moment (mathematics)」の詳細全文を読む スポンサード リンク
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