Answer Happy

Let Z0, Z1, Z2, . . . be a sequence of independent random
variables with expectation 0 and variance 1. For n ⩾ 1 define the
random variables
Xn = Zn + 1 3 Zn−1.
2 5. Let 20, 21, 22, ... be a sequence of independent random variables with expectation 0 and variance 1. For n > 1 define the random variables Xn = Zn + Zn-1. -: ) (a) Determine cov(Xn, Xn+k) for all n >1 and k > 0. [3 marks] (b) Let U and V be two random variables with mean 0 and non-zero variance. Define the function f:R [0,00) such that f(c) = E((U - cV)). Show that f is minimised at c= 0 if and only if U and V are uncorrelated. [2 marks] .