Stat 432: Homework 6
1. A model for social mobility. Suppose we can classify a man’s occupation as
professional, skilled labor or unskilled labor. Further suppose that of sons of
professionals, 80% become professionals, 15% become skilled laborers and 5%
become unskilled laborers. For sons of skilled laborers, 60% become skilled laborers,
25% become professionals and 15% become unskilled laborers. Finally, 50% of the
sons of unskilled laborers become unskilled laborers, 20% become professionals and
30% become skilled laborers. Assume every man has at least one son and form a
Markov chain by following the occupation of a randomly chosen son of a given
family through several generations.
a) Set up a matrix of one-step transition probabilities, e.g. the probability that a son
will a have a particular occupation given the father’s occupation.
b) Find the probability that a randomly chosen grandson of an unskilled laborer will
become a professional.
c) Suppose that the initial distribution of men’s occupations is 30% professionals,
50% skilled laborers and 20% unskilled laborers. What is the distribution of
men’s occupations after four generations?
2. In problem 1 we assumed that every man has at least one son. Instead assume that the
probability a man has at least one son is 0.8. Form a Markov chain with four states.
If a man has a son, the probability that this son goes into a particular occupation e.g.
professional, skilled laborer or unskilled laborer, is the same as in problem 1. If there
is no son, the process moves to state four which represents families whose male line
has died out.
a) Set up a matrix of one-step transition probabilities, e.g. the probability that a son
will a have a particular occupation given the father’s occupation.
b) Find the probability that a randomly chosen grandson of an unskilled laborer will
become a professional.
c) Suppose that the initial distribution of men’s occupations is 30% professionals,
50% skilled laborers and 20% unskilled laborers. What is the distribution of
men’s occupations after four generations?
3. Consider the Drunkard’s walk with five states; Bar(0), 1, 2, 3, and Home(4). Assume
the drunkard takes one step towards the bar with probability 0.6 and one step towards
home with probability 0.4.
a) Set up a matrix of one-step transition probabilities for this Markov chain.
b) Find the probability the drunkard will be home after 4 steps given he starts at state
2?
c) Suppose the drunkard is equally likely to start out at 1, 2 or 3, i.e.
1
pi 
i  1, 2, 3 . What is the probability the drunkard is at the Bar(0) after 4
3
steps?
1
d) Suppose the drunkard is equally likely to start out at 2 or 3 i.e. pi 
i  1, 2 .
2
What is the probability the drunkard is at the Bar(0) after 4 steps?
1