New England fall weather can be classified as sunny, cloudy, or rainy. A student conducted a detailed study of the weather pattern and came up with the following conclusion: Given that it is sunny on any given day, then on the following day it will be sunny again with probability 0.5, cloudy with probability 0.3 and rainy with probability 0.2. Given that it is cloudy on any given day, then on the following day it will be sunny with probability 0.4, cloudy again with probability 0.3 and rainy with probability 0.3. Finally, given that it is rainy on any given day, then on the following day it will be sunny with probability 0.2, cloudy with probability 0.5 and rainy again with probability 0.3.
a. Give the state-transition diagram of New England fall weather with the state "sunny" as state 1, the state "cloudy" as state 2, and the state "rainy" as state 3.
b. Using the same convention as in part (a), give the transition probability matrix of New England fall weather.
c. Given that it is sunny today, what is the probability that it will be sunny four days from now?
d. Determine the limiting-state probabilities of the weather.