Chapter 11.4, Problem 88AYU

To determine

To calculate: The value of $P^2$ where,

  $\begin{array}{l}\text{ Parent's income }\\ \begin{array}{ccccc}& L& M& H& \end{array}\\ P=\left[\begin{array}{ccc}0.4& 0.2& 0.1\\ 0.5& 0.6& 0.5\\ 0.1& 0.2& 0.4\end{array}\right]\begin{array}{c}L\\ M\\ H\end{array}\text{ Child's income}\end{array}$

Answer to Problem 88AYU

The value of $P^2$ is $\left[\begin{array}{ccc}0.27& 0.22& 0.18\\ 0.55& 0.56& 0.55\\ 0.18& 0.22& 0.27\end{array}\right]$ .

Explanation of Solution

Given information:

The left stochastic transition matrix is denote relative income of a child (low, medium or high) depends on the relative income of the child’s parents.

  $\begin{array}{l}\text{ Parent's income }\\ \begin{array}{ccccc}& L& M& H& \end{array}\\ P=\left[\begin{array}{ccc}0.4& 0.2& 0.1\\ 0.5& 0.6& 0.5\\ 0.1& 0.2& 0.4\end{array}\right]\begin{array}{c}L\\ M\\ H\end{array}\text{ Child's income}\end{array}$

Formula used:

To find the product AB of two matrices A and B, the number of column in matrix A must be equal to number of rows in matrix B.

Calculation:

Consider the left stochastic transition matrix is denote relative income of a child (low, medium or high) depends on the relative income of the child’s parents.

  $\begin{array}{l}\text{ Parent's income }\\ \begin{array}{ccccc}& L& M& H& \end{array}\\ P=\left[\begin{array}{ccc}0.4& 0.2& 0.1\\ 0.5& 0.6& 0.5\\ 0.1& 0.2& 0.4\end{array}\right]\begin{array}{c}L\\ M\\ H\end{array}\text{ Child's income}\end{array}$

Recall that to find the product AB of two matrices A and B, the number of column in matrix A must be equal to number of rows in matrix B.

Therefore, $P^2$ is,

  $\begin{array}{c}{P}^2=\left[\begin{array}{ccc}0.4& 0.2& 0.1\\ 0.5& 0.6& 0.5\\ 0.1& 0.2& 0.4\end{array}\right]\left[\begin{array}{ccc}0.4& 0.2& 0.1\\ 0.5& 0.6& 0.5\\ 0.1& 0.2& 0.4\end{array}\right]\\ =\left[\begin{array}{ccc}0.16+0.1+0.01& 0.08+0.12+0.02& 0.04+0.1+0.04\\ 0.2+0.25+0.01& 0.1+0.36+0.04& 0.05+0.3+0.2\\ 0.04+0.1+0.04& 0.02+0.12+0.08& 0.01+0.1+0.16\end{array}\right]\\ =\left[\begin{array}{ccc}0.27& 0.22& 0.18\\ 0.55& 0.56& 0.55\\ 0.18& 0.22& 0.27\end{array}\right]\end{array}$

The above matrix is interpreted as,

  $55\%$ of the children of low relative income parents will transition to medium level of income.

  $18\%$ of the children of low relative income parents will transition to high level of income.

  $27\%$ of the children of low relative income remain in same income level as their parents.

  $56\%$ of the children of medium relative income remain in same income level as their parents.

  $27\%$ of the children of high relative income remain in same income level as their parents.

Thus, the value of $P^2$ is $\left[\begin{array}{ccc}0.27& 0.22& 0.18\\ 0.55& 0.56& 0.55\\ 0.18& 0.22& 0.27\end{array}\right]$ .