Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 93the random column vector with components X 1 ,...,X n , and let the means ofX 1 ,...,X n be represented by the vector m X . A convenient representation oftheir variances and covariances is the covariance matrix, L, defined byL ˆ Ef…X m X †…X m X † T g; …4:34†where the superscript T denotes the matrix transpose. The n n matrix L hasa structure in which the diagonal elements are the variances and in which thenondiagonal elements are covariances. Specifically, it is given by23var…X 1 † cov…X 1 ; X 2 † ... cov…X 1 ; X n †cov…X 2 ; X 1 † var…X 2 † ... cov…X 2 ; X n †L ˆ6 . . .4. . . 75 : …4:35†cov…X n ; X 1 † cov…X n ; X 2 † ... var…X n †In the above ‘var’ reads ‘variance of’ and ‘cov’ reads ‘covariance of’. SincecovX i , X j ) ˆ covX j , X i ), the covariance matrix is always symmetrical.In closing, let us state (in Theorem 4.2) without proof an important resultwhich is a direct extension of Equation (4.28).Theorem 4. 2: if X 1 ,X 2 ,...,X n are mutually independent, thenEfg 1 …X 1 †g 2 …X 2 † ...g n …X n †g ˆ Efg 1 …X 1 †gEfg 2 …X 2 †g ...Efg n …X n †g;…4:36†where g j (X j ) is an arbitrary function of X j . It is assumed, of course, that allindicated expectations exist.4.4 MOMENTS OF SUMS OF RANDOM VARIABLESLet X 1 ,X 2 ,...,X n be n random variables. Their sum is also a random variable.In this section, we are interested in the moments of this sum in terms ofthose associated with X j , j ˆ 1, 2, . . . , n. These relations find applicationsin a large number of derivations to follow and in a variety of physicalsituations.ConsiderY ˆ X 1 ‡ X 2 ‡‡X n :…4:37†Let m j and 2 j denote the respective mean and variance of X j . Results 4.1–4.3are some of the important results concerning the mean and variance of Y.TLFeBOOK