Data have been accumulated on the heights of children relative to their parents. Suppose that the probabilities that a tall parent will have a tall, medium - height, or short child are 0.6, 0.2 and 0.2, respectively; the probabilities that a medium height parent will have a tall medium height or short child are 0.1, 0.7 and 0.2 respectively and the probabilities that a short parent will have a tall medium height or short child are 0.2, 0.4 and 0.4 respectively.
a. Write down the transition matrix for this Markov chain.
b. What is the probability that a short person will have a tall grandchild?
c. If 20% of the current population is tall, 50% is of medium height, and 30% is short what will the distribution be in three generations?
d. What proportion of the population will be tall, of medium height, and short in the long run?