Data have been accumulated on the heights ofchildren relative to their parents. Suppose

Chapter 3, Problem 3.7.9

(choose chapter or problem)

Data have been accumulated on the heights ofchildren relative to their parents. Suppose that theprobabilities that a tall parent will have a tall, mediumheight,or short child are 0.6, 0.2, and 0.2, respectively;the probabilities that a medium-height parent willhave a tall, medium-height, or short child are 0.1, 0.7,and 0.2, respectively; and the probabilities that a shortparent will have a tall, medium-height, or short childare 0.2, 0.4, and 0.4, respectively.(a) Write down the transition matrix for this Markovchain.(b) What is the probability that a short person willhave a tall grandchild?(c) If 20% of the current population is tall, 50% is ofmedium height, and 30% is short, what will thedistribution be in three generations?(d) If the data in part (c) do not change over time,what proportion of the population will be tall, ofmedium height, and short in the long run?