Pg 414 Q11

Assume that a man’s profession can be classified as professional, skilled laborer, or unskilled laborer. Assume that, of the sons of professional men, 80% become professionals themselves, 10% become skilled laborers, and 10% become unskilled laborers. For skilled laborers, 60% become skilled laborers, 20% become professionals, and 20% become unskilled laborers. For unskilled laborers, 50% become unskilled laborers, 25% become skilled laborers, and 25% become professionals. Assume that each man has at least one son and form a Markov chain by following the profession of a randomly chosen son of a given family through several generations.

Set up the matrix of transition probabilities. Find the probability that a randomly chosen grandson of an unskilled laborer becomes a professional.

The transition matrix is;

\[ \begin{bmatrix} & P & S & U \\ P & .8 & .1 & .1 \\ S & .2 & .6 & .2 \\ U & .25 & .25 & .5 \end{bmatrix} \]

To find the chance of a randomly chosen grandson of an unskilled labor being a professional, we’re asking for the following;

\[ P_{U,P}^{(2)} = P_{U,U}P_{U,P} + P_{U,S}P_{S,P} + P_{U,P}P_{P,P} \] \[ = .5*.25 + .25*.2 + .25*.8 \] \[ = .125 + .05 + .2 = .375 \]

Let’s use R to find steady state matrix of the grandson being a professional

library(expm)
## Warning: package 'expm' was built under R version 4.3.3
## Loading required package: Matrix
## 
## Attaching package: 'expm'
## The following object is masked from 'package:Matrix':
## 
##     expm
P <- matrix( c(.8, .2, .25, .1, .6, .25, .1, .2, .5), nrow=3, ncol=3)
row.names(P) <- c("P","S","U")
colnames(P) <- c("P","S","U")
P
##      P    S   U
## P 0.80 0.10 0.1
## S 0.20 0.60 0.2
## U 0.25 0.25 0.5
P_2 = P %*% P
P_2
##       P     S     U
## P 0.685 0.165 0.150
## S 0.330 0.430 0.240
## U 0.375 0.300 0.325