Problem 3. (4 points) The weather in a particular area behaves as follows: - The days can be sunny, cloudy, or rainy. ' Sunny days are followed by cloudy days 20% and by rainy days 10% of the time. ' Cloudy days are followed by sunny days 20% and by rainy days 60% of the time. ' Rainy days are followed by sunny days 40% and by cloudy days 20% of the time. Perform the following tasks by hand (do not use a computer): (a) Model the weather system as a Markov chain and give it as a graph. (b) Compute the steady-state probabilities using the j = 2:1 kpkj method. (c) Construct the transition matrix Q. Demonstrate that its A = 1 eigenvector gives the same steady-state probabilities as in (b). (If you construct Q row-wise, where qij gives the prob- ability to transition from i to j, you need the eigenvectors of QT.)