In Exercise 10, what is the expected number of generations until a short person has a tall descendant?
Suppose we want to know the average (or expected) number of steps it will take to go from state i to state j in a Markov chain. It can be shown that the following computation answers this question: Delete the jth row and the jth column of the transition matrix P to get a new matrix Q. (Keep the rows and columns of Q labeled as they were in P.) The expected number of steps from state i to state j is given by the sum of the entries in the column of (I -Q) -1 labeled i.
In Exercise 10
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.