1. Problem 11.16 Let the random variables X1,...,Xn be defined on a common probability space. Prove that n

var(X1 +···+Xn) =

i=1 n

var(Xi) + 2 i=1 n

j=i+1 cov(Xi, Xj).

Next, evaluate var( n i=1 ti Xi) in order to verify that n i=1 n

j=1 titjσij ≥ 0 for all real numbers t1,...,tn, where σij = cov(Xi, Xj). In other words, the covariance matrix C = (σij) is positive semi-definite.