For Xn, it wouldn't be a Markov Chain? Considering how the rules depends on the weather 2 days ago and yesterday? The process would then rely on Xn-1 right?

5. Consider the state space {rain, no rain} to describe the weather each day. Let {X,} be the stochastic process with this state space describing the weather on the nth day. Suppose the weather obeys the following rules: e If it has rained for two days straight, there is a 20% chance that the next day will be rainy. e If today is rainy but yesterday was not, there is a 50% chance that tomorrow will be fainy. ' e If yesterday was rainy but today is not, there is a 70% chance that tomorrow will be rainy. e If it has not rained for two days straight, there is a 90% chance that the next day will be rainy. (a) Is {X,} a Markov chain? Why or why not? (b) Now let {Yy.} be a different stochastic process defined as Y;, = (X, Xaz1), so that records the weather for the nth day as well as the (n1)st day. What is the state space for {V,}? (c) Show that {Y,} is a Markov chain and find its transition matrix