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a mathematical concept that plays an important role in mechanics and probability theory. If we have a system of point masses m1, m2, … (mi > 0) lying on a line and if their abscissas with respect to some origin O are equal to x1, x2, …, respectively, then the sum
x1km1 + x2km2 + … = Σ ixikmi
is called the kth order moment of the system about the point O. In mechanics the first-order moment is called the static moment and the second-order moment is called the moment of inertia. If in the expression for the moment all the abscissas are replaced by their absolute values, a quantity called the absolute moment is obtained. The point with abscissa (Σiximi)/(Σimi) is called the center of mass of a given system of masses. Moments computed about the center of mass are called central moments. The central, first-order moment for any system is equal to zero. Of all the moments of inertia, the central moment of inertia is the smallest. Chebyshev’s inequality states that the sum of the masses located at a distance greater than a from the point O does not exceed the system’s moment of inertia with respect to O divided by a2. If a mass distribution has a density f(x) ≧ 0, then the integral
is called the moment of order k, provided that it is absolutely convergent. In the case of an arbitrary mass distribution, the sums in the expressions for the moments are replaced by Stieltjes integrals; the Stieltjes integral first arose in this way. All the definitions and theorems mentioned above remain valid in this case.
In probability theory, the different possible values of a random quantity play the role of abscissas, and the corresponding probabilities take the place of the masses. The first-order moment is called the mathematical expectation value of the given random quantity, and the central, second-order moment is called its dispersion. The first-order moment is always the abscissa of the center, since the total mass is unity. Chebyshev’s inequality, referred to above, plays an extremely important role in probability theory. In mathematical statistics, moments usually function as basic, statistical quantities summarizing the characteristics of distributions.
The problem in mathematical analysis of the characterization of the properties of a function f(x) by the properties of the sequence of its moments
is called the problem of moments. This problem was first considered by P. L. Chebyshev in 1874 in connection with his studies in probability theory (the attempt to prove the central limit theorem). Later, powerful new methods of mathematical analysis were developed in the investigation of this problem.
REFERENCESChebyshev, P. L. Izbr. trudy. Moscow, 1955.
Markov, A. A. Izbr. trudy. Moscow, 1951.
Gnedenko, B. V. Kurs teorii veroiatnostei, 5th ed. Moscow, 1969.
Loève, M. Teoriia veroiatnostei. Moscow, 1962. (Translated from English.)