(See page 50 of the Cryer and Chan text, Exercise 3.2)

Mark, rho_k is the autocorrelation function at lag k, which is given to us to be gamma_k/gamma₀.  Gamma_k is the autocovariance function at lag k (which should be written gamma_0,k, but is simplfied for our convenience), Cov(Y_t,Y_t-k). 

Thus, rho₁ = gamma₁/gamma₀ = Cov(Y_t,Y_t-1)/Cov(Y_t,Y_t-0) = Cov(Y_t,Y_t-1)/Var(Y_t).

For problem 2, we are asked to find Var(Y-bar) of an MA(1) process: Y_t = mu + e_t + .5e_t-1 with 50 observations (so we can't use the estimate formula, as n should be greater than 50), and Var(e_t) = 1.  We will use the formula on page 28: Var(Y-bar) = (gamma₀/n)[1+2(1-1/n)(rho₁)], this is a reduced version of the summation formula above on the same page because an MA(1) process has rho_k=0 for k>1.

First, find the variance of Y_t using the rules of variance and assume the independence of the e's:  Var(Y_t) = Var(mu + e_t+ .5e_t-1) = Var(mu) + Var(e_t) + Var(e_t-1) = 0 + Var(e_t) + Var(e_t) = gamma₀.

Next, in order to find rho₁, we must find gamma₁ = Cov(Y_t,Y_t-1) = Cov(e_t+.5e_t-1,e_t-1+.5e_t-2) = Cov(.5e_t-1,e_t-1) = .5Var(e_t-1).  You now have enough information to complete the given formula for Var(Y-bar).  Let me know if you have further questions.

RDH