Problem 3. Suppose that we characterize the weather in a certain rain forest by only three states: State 1 is cloudy, state 2 is rainy, and state 3 is sunny. The daily weather then forms a markov chain. The probability-transition diagram, as observed over a long period, is shown in Figure 11.1.2. Figure 11.1.2. Probability-transition diagram of day-to-day weather changes in a rain forest (Problem 3). By this we imply the probability transition matrix, 19.0: + 1) .6 .3 .2 pt(k) [174k + 1)] = [.3 .7 .5][p2(k)] . 1230: + 1) .1 0.3 193(k) Show that a stationary probability mass function exists (as k> 6) for this system, with 1'1 = .42 ,p2 = .52', p3 = .06. ' Given that a certain day is sunny (p3 = 1 ,1), = 192 = 0) , predict the weather for the next few days