Assume that each day is either sunny or rainy, which depends on the previous 2 days' weather as follows: Assume that if today is sunny and/or yesterday was sunny then tomorrow will again be sunny with a fixed probability 0.8. Assume that if today is rainy and yesterday was rainy then tomorrow will again be rainy with a fixed probability 0.5. Let (X_n) be the process that keeps track of the weather on day n. This process is not a discrete-time Markov chain because each day's weather depends on the previous two days' weather but it is possible to analyze the process:

1. Find the transition matrix of the process (Y_n) where Y_N = (X_n, X_n+1).

2. Deduce the fraction of days which are sunny in the long run.

Assume now that the number of car accidents is Poisson distributed with parameter two when it is sunny but three when it is rainy.

3.Find the fraction of days with no car accident in the long run.

Assume now that each day is either sunny or rainy and that today's weather is the same as yesterday's weather with a probability p (0,1).

1. Compute the fraction of days that in the long run are sunny.

2. Prove more generally that the probability that the weather on n days will be the same as today's weather is equal to 0.5+(0.5)*(p-q)^n where q = 1-p (0,1).