for this result. 7. A machine consists of two parts that fail and are repaired in- dependently. A working part fails during any given day with probability a. A part that is not working is repaired by the next day with probability b. Let X be the number of working parts in day n. (a) Show that X is a three-state Markov chain and give its one-step transition probability matrix P. (b) Show that the steady state pmf 7 is binomial with param- eter p = b/(a+b). (c) What do you expect is the steady state pmf for a machine that consists of m parts? 3. (7 points) For each of the following ARMA prcoesses, (1) Simulate a time series with 60 observations. (2) Make a sample correlogram using the simulated data. (3) Generate and plot the theoretical autocorrelation functions up to 15 lags. (a) MA(1) with 0 = 0.6 (b) MA(1) with 0 = -0.6 01 (c) MA(2) with 0 = 0.4 and 02 = 0.2 (d) AR(1) with p = 0.4 (e) AR(1) with = -0.4 (f) AR(2) with = 0.5 and 2 = -0.9 1 (g) ARMA (1,1) with = 0.7 and 0 = 0.4