Suppose that xt 1 N(xt 10s) and xt is generated by xt = ztxt 1 +(1 zt) t where t N(0s) and zt is binary with Pr(zt = 1) = a and with xt 1 zt t being mutually independent. Here sa are known.

(a) What is the distribution of xt?

(b) What is the correlation C(xt xt 1)?

(c) Simulate and graph realizations of such a process x1:T for T =1,000 with s = 1 and for each of a = 09099 Can you suggest a context in which such a model might be useful?