Moment (mathematics)

In mathematics, a moment is a specific quantitative measure of the shape of a function. It is used in both mechanics and statistics. If the function represents physical density, 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 function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. In mathematics, a moment is a specific quantitative measure of the shape of a function. It is used in both mechanics and statistics. If the function represents physical density, 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 function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. 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). The n-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 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 n-th inverse moment about zero is E ⁡ [ X − n ] {displaystyle operatorname {E} left} and the n-th logarithmic moment about zero is E ⁡ [ ln n ⁡ ( X ) ] . {displaystyle operatorname {E} left.} The n-th moment about zero of a probability density function f(x) is the expected value of Xn and is called a raw moment or crude moment. The moments about its mean μ 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 n-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 n-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 E is the expectation operator or mean.