Markov Models Problems
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Q151. A cinema has a marquee with lots and lots of light bulbs. In any given week 1% of the light bulbs burn out. Unfortunately, between being busy and being sloppy, replacement is a little bit sporadic. Of all the bulbs that are burnt out, about 95% get replaced each week. Draw the state diagram for this system.

Q152. Translate this description into a state diagram. A population consists of people who play it safe, and daredevils. From year to year, most (97%) safe-players stay that way, but 2% turn into daredevils. About 1% of the safe-players die each year. By contrast, 10% of daredevils die each year and another 10%, seeing that, switch to playing it safe. All the other daredevils stick with the program.

Label each of the states in as transient or absorbing.

Q153. A criminologist and an activist decide to collaborate on a project designed to reduce prison population. In the spirit of starting simple, they identify 4 states in which people can find themselves: never imprisoned; incarcerated; on parole; post-parole. The period of time in their analysis will be one year. Suppose 70% of the population has never been incarcerated. Each year 2% of these people are imprisoned. Of those currently incarcerated, 20% are released each year onto parole. Average parole is 5 years so that a person on parole has a 20% chance of finishing parole. Those on parole have a 10% chance of finding themselves back in prison in any given year. Individuals who are post parole have a 4% chance of returning to prison in any given year. Draw a state diagram and matrix representing this information.

Q154. Suppose a given housing market has a 10% turnover rate each year. How many houses will typically be in the first, second, third, etc. year of their mortgage at any given time? Assume 10 year mortgages to keep things simple. Draw the diagram that would be the first step in solving this problem.

Q155. Suppose 25% of the mortgages written in the first years of this century were subprime (meaning the borrowers were not very credit-worthy) and all were 5 year adjustable such that after the fifth year the monthly payments would go way up. In the market in question there is approximately 5% turnover housing each year. The housing stock in the market consists of one million units. Research has shown that 33 1/3% of subprime adjustable mortgages go into default under current conditions when they go past their five year mark (and these conditions are expected to continue for some time) when they adjust.

Q156. A state corrections system has established a new drug treatment facility for first offenders. The center has a capacity of 1000.

Inmates may leave the facility in either of two ways. In any period, there is a 10 percent probability that an inmate will be judged rehabilitated, in which case s/he will be released at the beginning of the following period.. There is also a 5 percent chance that an inmate will escape during each period. Rehabilitated addicts have a 20 percent chance of relapsing in each period; escapees have a 10 percent chance of being recaptured each period. Both recidivists and recaptured escapees are returned to the facility and have priority over new offenders.

Questions

If it operates at full capacity, how many of the original inmates will be resident at the facility 10 periods later?
How many new offenders can be admitted during each of the next 2 periods?
What happens if we modify the model to allow for a small possibility of death or a change in the probability of relapse?

Q157. Sketch the transition matrix that corresponds to the following diagram

Q158. Suppose the following statements are true about the local housing market.

1. On a month to month basis, 90% of mortgage payments are on time, 10% are late or missed.
2. Of all the late/missed payments, 25% are back on track the following month. 65% are late again. 10% go into default.
3. Of all mortgages in default in a given month, 20% have a work-out and return to good standing. 70% remain in default and 10% move into foreclosure.
4. Of all houses in foreclosure each month, the banks manage to get 20% back on the market and resold.

Draw the transition diagram and write out the transition matrix.

Q159. Implement the model from problem 158 using this Excel spreadsheet.

Q187. The diagram below represents a candidate's shifting weekly position on abortion. Treating this as a Markov model (where each transition is independent of previous sequence of states), show us what the transition matrix would look like. What do we call the state that would represent a candidate's final position on an issue? Using this diagram, what prediction can you make about what this candidate's position would be if he were to be elected?

Q328. Have a look at this recent release from Bureau of Labor Statistics (BLS). The data separates those without a job into unemployed but "in the labor force" and "marginally attached to the labor force" and a subset of these called "discouraged" - the former would like to work but have not looked in the last four weeks and so are not counted as unemployed. The latter are not actively looking for work having given up on the idea that its possible to find. These groups are not included in the denominator when the unemployment rate is calculated. The simple version of the unemployment rate is, then,

(1)
\begin{align} UR = \frac {Unemployed} {Employed + Unemployed} \end{align}

Some recent op-eds have counseled caution about optimism that the overall unemployment rate has been going down because it might reflect growth in the number of people no longer looking for work. We'll think about that with a Markov model. We'll simplify the states a worker can be in:

• employed (E)
• short term unemployed - 14 weeks or less (US)
• long term unemployed - over 14 weeks (LS)
• Marginally attached to the labor force - no longer looking for a job (MALF)

Let's construct a simplified Markov model of unemployment based on transition rates shown here:

If the unemployment rate is calculated as the ratio of those who are short term unemployed (US) plus those who are long term unemployed (UL) to the total labor force (E + US + UL), how would things evolve over the next twelve months if the starting numbers are these:

What will the unemployment rate be? Even if it is agreed that getting unemployment to near 6% is a policy goal, are there reasons the results might not be a cause for celebration?

Create a chart showing changes over the next 12 months. Suggestion: plot total employment (E) on secondary axis since it's such a large number. In the alternative, put it on a separate chart.

Excel worksheet here

page revision: 8, last edited: 13 Oct 2012 18:52