Moment (mathematics)
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- See also moment (physics).
The concept of moment in mathematics evolved from the concept of moment in physics. The nth moment of a real-valued function f(x) of a real variable is
- <math>\mu'_n=\int_{-\infty}^\infty x^n\,f(x)\,dx.<math>
The problem of moments seeks characterizations of sequences { μ′n : n = 1, 2, 3, ... } that are sequences of moments of some function f.
If (lower-case) f is a probability density function, then the value integral above is called the nth moment of the probability distribution. More generally, if (capital) F is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the nth moment of the probability distribution is given by the Riemann-Stieltjes integral
- <math>E(X^n)=\int_{-\infty}^\infty x^n\,dF(x),<math>
where X is a random variable that has this distribution.
The nth central moment of the probability distribution of a random variable X is
- <math>\mu_n=E((X-\mu_1')^n).<math>
The second central moment is the variance.
The central moments are clearly translation-invariant, i.e., the nth central moment of X is the same as that of X + c for any constant c (in this context "constant" means a non-random quantity).
The first moment and the second and third central moments are linear in the sense that if X and Y are independent random variables then
- <math>\mu_1(X+Y)=\mu_1(X)+\mu_1(Y)<math>
and
- <math>\operatorname{var}(X+Y)=\operatorname{var}(X)+\operatorname{var}(Y)<math>
and
- <math>\mu_3(X+Y)=\mu_3(X)+\mu_3(Y)<math>
(independence is not needed for the first of these three identities; for the second it can be weakened to uncorrelatedness).
The central moments beyond the third lack this linearity; in that respect they differ from the cumulants (the first three cumulants are the same as the first moment and the second and third central moments; the higher cumulants have a more complicated relationship with the central moments).
Like the cumulants, the factorial moments of a probability distribution are also polynomial functions of the moments.
