Moments of a function in mathematics are certain quantitative measures related to the shape of the function's graph. For example, if the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from 0 to ∞) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the moments of random variables.
Significance of the moments The nth raw moment (i.e., moment about zero) of a random variable X {\displaystyle X} with density function f ( x ) {\displaystyle f(x)} is defined by μ n ′ = ⟨ X n ⟩ = d e f { ∑ i x i n f ( x i ) , discrete distribution ∫ x n f ( x ) d x , continuous distribution {\displaystyle \mu '_{n}=\langle X^{n}\rangle ~{\overset {\mathrm {def} }{=}}~{\begin{cases}\sum _{i}x_{i}^{n}f(x_{i}),&{\text{discrete distribution}}\\[1.2ex]\int x^{n}f(x)\,dx,&{\text{continuous distribution}}\end{cases}}} The nth moment of a real-valued continuous random variable with density function f ( x ) {\displaystyle f(x)} about a value c {\displaystyle c} is the integral μ n = ∫ − ∞ ∞ ( x − c ) n f ( x ) d x . {\displaystyle \mu _{n}=\int _{-\infty }^{\infty }(x-c)^{n}\,f(x)\,\mathrm {d} x.}
It is possible to define moments for random variables in a more general fashion than moments for real-valued functions – see moments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with c = 0 {\displaystyle c=0} . For the second and higher moments, the central moment (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 nth inverse moment about zero is E [ X − n ] {\displaystyle \operatorname {E} \left[X^{-n}\right]} and the nth logarithmic moment about zero is E [ ln n ( X ) ] . {\displaystyle \operatorname {E} \left[\ln ^{n}(X)\right].}
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