This is the problem: There are 2n − 1 independent random variables X1, ?2, ... , ?2?−1. The expectation E(Xi) is μ for all i = 1,...,2n−1. The variance Var(Xi) is σ2 for all i = 1,...,2n−1. Let Y=X1 +?2 +⋯+?? . W= Xn+1 + Xn+2 +⋯+ X2n−1 .
Derive the expression for the covariance Cov(Y, W) and the correlation ρYW.
My attempted solution: The covariance of (Y,W) = Cov (X1+X2+...Xn,Xn+1+Xn+2...X2n-1). This expands to the summation of the covariances of each pair of variables, i.e. - Cov(X1,Xn+1)+Cov(X2,Xn+1)+...Cov(Xn,Xn+1). But each of these covariances = 0 because they are all independent (according to the question). So Cov(Y,W)=0.
Corr(Y,W)=0 as well because Corr(Y,W)=Cov(Y,W)/((Var(Y)Var(W))^.5]
I'm just checking to see if this sounds reasonable. Thanks for any input!!
Derive the expression for the covariance Cov(Y, W) and the correlation ρYW.
My attempted solution: The covariance of (Y,W) = Cov (X1+X2+...Xn,Xn+1+Xn+2...X2n-1). This expands to the summation of the covariances of each pair of variables, i.e. - Cov(X1,Xn+1)+Cov(X2,Xn+1)+...Cov(Xn,Xn+1). But each of these covariances = 0 because they are all independent (according to the question). So Cov(Y,W)=0.
Corr(Y,W)=0 as well because Corr(Y,W)=Cov(Y,W)/((Var(Y)Var(W))^.5]
I'm just checking to see if this sounds reasonable. Thanks for any input!!