tailieunhanh - Handbook of mathematics for engineers and scienteists part 157

Handbook of mathematics for engineers and scienteists part 157. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 1060 Probability Theory Random variables Xi and X2 are said to be independent if the relation P Xi e Si X2e S2 P Xi e Si P X2e S2 holds for any measurable sets Si and S2. Theorem 1. Random variables Xi and X2 are independent if and only if Fxi x2 xi X2 Fxi xi Fx2 x2 . Theorem 2. Random variables Xi and X2 are independent if and only if the characteristic function of the bivariate random variable Xi X2 is equal to the product of the characteristic functions of Xi and X2 fxi x2 xi X2 fxi xi fx2 x2 . . Numerical characteristics of bivariate random variables. The expectation of a function g X1 X2 of a bivariate random variable X1 X2 is defined as the expression computed by the formula EEg x1i X2j pij in the discrete case if 0Oif œ g x1 x2 p x1 x2 dx1 dx2 in the continuous case J-œ J -œ if these expressions exist in the sense of absolute convergence otherwise one says that E g X1 X2 does not exist. The moment of order r1 r2 of a two-dimensional random variable X1 X2 about a point ai af is defined as the expectation E X1 - a1 r1 X2 - a2 r2 . If a1 a2 0 then the moment of order r1 r2 of a two-dimensional random variable Xi X2 is called simply the moment or the initial moment. The initial moment of order r1 r2 is usually denoted by arir2 . ar1 r2 E XpX 2 . The first initial moments are the expectations of the random variables X1 and X2 . 1 0 E X1X0 E X1 and 0 1 E X0X2 E X2 . The point E XJ E X2 on the OXY-plane characterizes the position of the random point X1 X2 this position spreads about the point E X1 E X2 . Obviously the first central moments are zero. The second initial moments are given by the formulas 2 0 2 X1 0 2 2 X2 1 1 E X1X2 . If a1 E X1 and a2 E X2 then the moment of order r1 r2 of the bivariate random variable X1 X2 is called the central moment. The central moment of order r1 r2 is usually denoted by pr1 r2 . pr1 r2 E X1 - E X1 r1 X2 - E X2 r2 . The second central moments are of special interest and have special names and .