Based on accumulated data concerning the heights of male children relative to their fathers, it has been determined that the probabilities that a tall man will have a tall medium-height or short child are 0.6, 0.2 and 0.2 respectively. The probabilities that a man of medium height will have a tall, medium-height or short child are 0.1, 0.7 and 0.2 respectively, and the probabilities that a short man 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 man will have a tall grandson?
(c) If 20% of the current male population is tall, 50% is of medium height and 30% is short, what will the distribution be in three generations?
(d) According to this model, what will the distribution be in the long run?
(a) The transition matrix is the matrix whose entries `a_(i,j) ` are the probabilities that an object in state i moves to state j. The transition matrix is:
`T=[[.6,.2,.2],[.1,.7,.2],[.2,.4,.4]] ` (Note that the .7 in entry 2,2 is the probability that a medium height man will have a medium height son.) Here the first row is for tall men and the column entries are tall, medium, then short.
(b) To find the probability that a short man will have a tall grandson we look at the square of the transition matrix. (Multiplying a state vector by the transition matrix once moves through one time step -- multiplying by the square is multiplying twice and moves through 2 time steps.)
Thus the probability is .24
(c) To find the distribution in three generations, we take the initial state vector times the cube of the transition matrix.
(d) The distribution in the long run is the steady state vector. Since all entries in T are nonzero there is a unique steady state vector.
We could solve the following matrix equation:
`[[x,y,z]]*T=[[x,y,z]] ` where `[[x,y,z]] ` is the steady state vector. This is equivalent to solving the following linear system:
Or we could evaluate `T^n ` for "large" values of n. In either case we find the steady state vector to be `[[.25,.5,.25]] `
Thus the distribution in the long run would be 25% tall, 50% medium and 25% short.
The algebraic way is to solve the system.
If you evaluate T^n for "large" values of n, you will see that the result gets closer and closer to [.25 .5 .25]. Try T^25 and T^30 to see what happens. You can do this with any transition matrix -- occasionally you will have to use fairly large values for n (say in the hundreds or even larger) but you will see that the entries eventually stay the same.