Problem 4: A study determined that, in a certain region, the occupation of a person, as an adult, depends upon the occupation of her/his parent, and is given by the following probabilities: If the parent was a professional (P), the probability that the person will be a professional (P) is 0.8, a farmer (F) is 0.1, and a laborer (L) is 0.1; If the parent was a farmer (F), the probability that the person will be a professional (P) is 0.3, a farmer (F) is 0.5, and a laborer (L) is 0.2; If the parent was a laborer (L), the probability that the person will be a professional (P) is 0.2, a farmer (F) is 0.2, and a laborer (L) is 0.6. It is also known that, in that region, the occupation of a person only depends on the occupation of his/her parent, i.e., it does not depend on the occupation of his grandparent. (a) Draw a state diagram for the Markov chain capturing the evolution of occupations over one generation (there will be 3 states: P, F and L). (b) Write down the stochastic matrix of the Markov chain. (c) Assume that, at the beginning of an observation period, there are only farmers (F) in the region. What is the probability to have professionals (P) and laborers (L) in three generations in the future? (d) In long term, what is the distributions of the three occupation over the population in that region? (e) Calculate the eigenvalues and eigenvectors of the stochastic matrix of the Markov chain and discuss the connec- tion with the results you got in (d).