Answer Happy

1. Let Y, t e N, be a sequence of independent, identically distributed ran- dom variables on a given probability space (12, F, P) such that E(Y) = 0 and Var(Y) = 1. (i) Define X] :=Y, and for t > 2 X :=Y1-1+Y. Compute E(Xi) and Cov(X2, Xx-x) for t e N and for s = 0, 1, 2,...,t - 1. (30 Marks) (ii) Define 2 = Y, and 2:= 24-1+Y. > 2. Prove that for arbitrarily fixed t e N and all s = 0,1,2,....t-1 E(Z) = 0, and Cov(Z1, Z-s) = t-s
2. Given a probability space (12, F,P). A time series process X.t e Z is called a weakly stationary process, if (1) E(X) = constant, Vt e Z; and (2) Cov(X4, Xx) = plt - s), Vt, s € Z, for some function p: Z R. Let the times series process X = {X/(w): € NW EN} be defined as X/(w) := sin(t +U(w)), te NW EN where U : N2 + (-27,0) be a uniformly distributed random variable on (2, F,P). Prove that X is a weakly stationary process. [20 Marks] {Hint: use the identity: sin(A) sin(B) = } [cos(A - B) - cos(A + B)].}
3. Let X, te N be a sequence of independent, identically distributed random variables on a given probability space (12,F,P). Set Y := X1, Y := Xt-1 + X7, t > 2. Show that the time series process Y = {Y/(w)}tenweg is a purely indetermin- istic process. [20 Marks)