tailieunhanh - Handbook of mathematics for engineers and scienteists part 158

Handbook of mathematics for engineers and scienteists part 158. 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. | . Random Variables and Their Characteristics 1067 For continuous random variables f f j i j j P x dx. The inversion formula for a continuous distribution has the form 1 i- x L -L j j f t dt where dt dti . dtn. If the initial moments of a random variable X exist then E Xr1 .Xnn i n 1 rj drl - rn f t dti . . . dtn 1 tn 0 The characteristic function corresponding to the m-dimensional marginal distribution of m out of n variables X1 . Xn can be obtained from the characteristic function if the variables tj corresponding to the random variables that are not contained in the m-dimensional marginal distribution are replaced by zeros. Continuity theorem for characteristic functions. The weak convergence Fxn F of a sequence of distribution functions F1 x F2 x . is equivalent to the uniform convergence fn t - f t of characteristic functions on each finite interval. . Independency of random variables. Random variables Xi . Xn are said to be independent if the events Xi e Si . Xn 6 Sn are independent for any measurable sets Si . Sn. For this it is necessary and sufficient that n P Xie Si . Xn Sn n p X Sk . k 1 Relation is equivalent to one of the following three 1. In the general case for any x e Rn n Fx x n 1 x Xk . k 1 2. For absolutely continuous distributions for any x e Rn except possibly for a set of measure zero n px x n pxk xk . k 1 3. For discrete distributions for any x e Rn n P Xi Xi . Xn Xn J P Xk Xk . k 1 The joint distribution of independent random variables is uniquely determined by their individual distributions. Independent random variables are uncorrelated but the converse is not true in general. Random variables Xi . Xn are independent if and only if the characteristic function of the multivariate random variable X is equal to the product of the characteristic functions of the random variables Xi . Xn. 1068 Probability Theory . Limit Theorems . Convergence of Random Variables . Convergence in .