Problem 2.1 Let X1, X2, X3 be random variables with zero-mean and unit variance: E(Xi) = 0 and var(Xi) = 1. Assume that the covariance between any two of these variables is ρ. Let us define Z1 = X1 − X2, Z2 = X2 − X3, Z3 = 2 Let Z = (Z1, Z2, Z3) ∈ R 3 . (a) Find the covariance matrix of Z, that is, cov(Z). What is the correlation coefficient between Z1 and Z2? Hint: Constant shifts do not affect covariances. (b) Is cov(Z) positive semi-definite? Is it possible to find a nonzero nonrandom a ∈ R 3 such that var(a TZ) = 0? (c) Find the variance of Z1 − Z2. (d) Let Y = 5Z + (1, 2, 3) = (5Z1 + 1, 5Z2 + 2, 5Z2 + 3) ∈ R 3 . Find cov(Y ).
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