can you help me understand theorem 9.1?

9.1 Expected Values of Sums The expected value of a sum of any random variables is the sum of the expected values. The variance of the sum of any random variable is the sum of all the covariances. The variance of the sum of independent random variables is the sum of the variances. 306 9.1 EXPECTED VALUES OF SUMS 307 The theorems of Section 5.7 can be generalized in a straightforward manner to describe expected values and variances of sums of more than two random variables. Theorem 9.1 For any set of random variables Xj...., Xn, the sum W, = Xi, +---+X, has expected value B[W,] =B[X1] +B[X2] +--+ BUX). Proof We prove this theorem by induction on n. In Theorem 5.11, we proved E/W2] = E[X1] + E[X2]. Now we assume E[W,,-1] = E[|Xi] + --- +E[X,_:i]. Notice that W, = Wr-1+X,. Since W,, isasum of the two random variables W,,-); and X,. we know that E(W,,] = E[Wp-1] + ELX,] = E[Xi] + --- + ELX,_-1] + ELX]. Keep in mind that the expected value of the sum equals the sum of the expected ralues whether or not Xj..... X, are independent. For the variance of W,,. we have the generalization of Theorem 5.12