It is raining in the Land of Oz. Determine a tree and a tree measure for the next three days' weather. Find [math]\mat {w}^{(1)}, \mat{w}^{(2)},[/math] and [math] \mat {w}^{(3)}[/math] and compare with the results obtained from [math] \mat {P},~\mat{P}^2,[/math] and [math] \mat {P}^3[/math].

In Example, let [math]a = 0[/math] and [math]b = 1/2[/math]. Find [math] \mat {P},~ \mat {P}^2,[/math] and [math] \mat {P}^3.[/math] What would [math] \mat {P}^n[/math] be? What happens to [math] \mat {P}^n[/math] as [math]n[/math] tends to infinity? Interpret this result.

In Example, find [math]\mat{P}[/math], [math] \mat {P}^2,[/math]and [math] \mat {P}^3.[/math] What is [math] \mat {P}^n[/math]?

For Example, find the probability that the grandson of a man from Harvard went to Harvard.

In Example, find the probability that the grandson of a man from Harvard went to Harvard.

In Example, assume that we start with a hybrid bred to a hybrid. Find [math] \mat {u}^{(1)},[/math] [math] \mat {u}^{(2)},[/math] and [math] \mat {u}^{(3)}.[/math]

What would [math] \mat {u}^{(n)}[/math] be?

Find the matrices [math]\mat{ P}^2,~\mat {P}^3,~\mat {P}^4,[/math] and [math] \mat {P}^n[/math] for the Markov chain determined by the transition matrix [math] \mat {P} = \pmatrix{ 1 & 0 \cr 0 & 1 \cr}[/math]. Do the same for the transition matrix [math] \mat {P} = \pmatrix{ 0 & 1 \cr 1 & 0 \cr}[/math]. Interpret what happens in each of these processes.

A certain calculating machine uses only the digits 0 and 1. It is supposed to transmit one of these digits through several stages. However, at every stage, there is a probability [math]p[/math] that the digit that enters this stage will be changed when it leaves and a probability [math]q = 1 - p[/math] that it won't. Form a Markov chain to represent the process of transmission by taking as states the digits 0 and 1. What is the matrix of transition probabilities?

For the Markov chain in Exercise, draw a tree and assign a tree measure assuming that the process begins in state 0 and moves through two stages of transmission. What is the probability that the machine, after two stages, produces the digit 0 (i.e., the correct digit)? What is the probability that the machine never changed the digit from 0? Now let [math]p = .1[/math]. Using the program MatrixPowers, compute the 100th power of the transition matrix. Interpret the entries of this matrix. Repeat this with [math]p = .2[/math]. Why do the 100th powers appear to be the same?

Modify the program MatrixPowers so that it prints out the average [math] \mat {A}_n[/math] of the powers [math]\mat {P}^n[/math], for [math]n = 1[/math] to [math]N[/math].

Try your program on the Land of Oz example and compare [math]\mat {A}_n[/math] and [math]\mat {P}^n.[/math]