Chapter 15, Problem 1RE

To determine

The S₁ and S₂, if $S_0=\left[\begin{array}{cc}0.3& 0.7\end{array}\right]\text{ and }P=\left[\begin{array}{cc}0.6& 0.4\\ 0.2& 0.8\end{array}\right]$ and explain the representation of S₁ and S₂.

Answer to Problem 1RE

The S₁ matrix is $\underset{\_}{\left[\begin{array}{cc}0.32& 0.68\end{array}\right]}$.

The S₂ matrix is $\underset{\_}{\left[\begin{array}{cc}0.328& 0.672\end{array}\right]}$.

Explanation of Solution

The initial state matrix S₀ and the transition matrix P is given below:

$S_0=\left[\begin{array}{cc}0.3& 0.7\end{array}\right]\text{ and }P=\left[\begin{array}{cc}0.6& 0.4\\ 0.2& 0.8\end{array}\right]$

If P is the transition matrix and S₀ is an initial-state matrix for a Markov chain, then the kth-state matrix is given by $S_k=S_0{P}^k$.

The entry in the ith row and jth column of $P^k$ indicates the probability of the system moving from the ith state to the jth state in k observations or trials.

The sum of the entries in each row of $P^k$ is 1.

Therefore,

$\begin{array}{c}{S}_1=S_0{P}^1\\ =\left[\begin{array}{cc}0.3& 0.7\end{array}\right]\left[\begin{array}{cc}0.6& 0.4\\ 0.2& 0.8\end{array}\right]\end{array}$

In order to multiply matrices,

Make sure that the number of columns in the first matrix equals the number of rows in the second matrix.

Multiply the elements of each row of the first matrix by the elements of each column in the second matrix.

$\begin{array}{c}{S}_1=\left[\begin{array}{cc}0.3& 0.7\end{array}\right]\left[\begin{array}{cc}0.6& 0.4\\ 0.2& 0.8\end{array}\right]\\ =\left[\begin{array}{cc}\text{0}\text{.3}\times \text{0}\text{.6+0}\text{.7}\times 0.2& \text{0}\text{.3}\times \text{0}\text{.4+0}\text{.7}\times 0.8\end{array}\right]\\ =\left[\begin{array}{cc}0.32& 0.68\end{array}\right]\end{array}$

Hence, the S₁ matrix is $\underset{\_}{\left[\begin{array}{cc}0.32& 0.68\end{array}\right]}$.

This matrix gives us the probabilities of a randomly chosen person being in state A or B on the first trial after the start of the value.

Now see that the A's value has increased from 30% to 32 %.

Now calculate $S_2=S_0{P}^2$ as follows.

Calculate the $P^2$ matrix:

$\begin{array}{c}{P}^2=\left[\begin{array}{cc}0.6& 0.4\\ 0.2& 0.8\end{array}\right]\left[\begin{array}{cc}0.6& 0.4\\ 0.2& 0.8\end{array}\right]\\ =\left[\begin{array}{cc}\text{0}\text{.6}\times 0.6+0.4\times 0.2& \text{0}\text{.6}\times \text{0}\text{.4+0}\text{.4}\times 0.8\\ \text{0}\text{.2}\times 0.6+0.8\times 0.2& \text{0}\text{.2}\times \text{0}\text{.4+0}\text{.8}\times 0.8\end{array}\right]\\ =\left[\begin{array}{cc}0.44& 0.56\\ 0.28& 0.72\end{array}\right]\end{array}$

Therefore,

$\begin{array}{c}{S}_2=\left[\begin{array}{cc}0.3& 0.7\end{array}\right]\left[\begin{array}{cc}0.44& 0.56\\ 0.28& 0.72\end{array}\right]\\ =\left[\begin{array}{cc}0.3\times 0.44+0.7\times 0.28& 0.3\times 0.56+0.7\times 0.72\end{array}\right]\\ =\left[\begin{array}{cc}0.328& 0.672\end{array}\right]\end{array}$

Hence, the S₂ matrix is $\underset{\_}{\left[\begin{array}{cc}0.328& 0.672\end{array}\right]}$.

This matrix gives us the probabilities of a randomly chosen person being in state A or B on the second trial after the start of the value.

Now see that the A's value has increased from 30% to 32.80 %.