## 2001 Term Project

### Due: Friday, December 14, 2001 @ 17:00

#### What you need to turn in to complete this project:

- A précis of your work—include a description of the algorithm (s) you use and the structure of your program. This is the place to discuss any difficulties you might have encountered.
- Your program, with LOTS of comments and dcoumentation.
- Output from representative runs.
- Your results, in tabular form, including any (optional) graphical representations you might wish to include. Since this is Monte Carlo, estimate the statistical uncertainty of your answers.

#### The problem:

Pretend you are a criminal who must choose among different types of crimes. The justice system provides a probability *q*_{k} that if you commit the k'th type of crime you will be caught and punished. Each kind of crime has a distribution of sentence length,

*p*_{k}(*J*)=4*J*exp[–2*J*/*j*_{
k}]/(*j*_{k})^{2}

where *j*_{k} is the average sentence for each kind of crime. That is, if you are caught for crime k you will receive a sentence of length *J* years with probability *p*_{k}(*J*)*dJ*. On the other hand, you receive a payment *m*_{k} for each successful crime.

**Capture and sentencing parameters**
Category of crime |
**q** |
**j** (years) |
**m** (dollars) |

larceny |
0.05 |
2 |
50 |

robbery |
0.1 |
4 |
100 |

burglary |
0.15 |
6 |
500 |

arson |
0.25 |
10 |
1000 |

In a time *dt* you can commit *R*_{k}*dt* crimes of type k, gaining profit

*dP*_{k}=(1-*q*_{k})*m*_{k}
*R*_{k}*dt*.

Of course you could go to jail, in which case your personal clock is advanced by *J* years.

Your task is to find values of *R*_{k} that will yield you at least $16,000 total profit per year and that will maximize your lifetime earnings (by minimizing the total time you spend in jail).
What would your answer be if you had a legitimate income of $30,000/yr?

#### Suggestions

After some discussions with students who asked about the project, I decided I had better offer some suggestions as to how to proceed.
- First write a subroutine to trace a single criminal's working life. Take his (her?) career to begin at age 16 and to end at age 65, giving 50 years of gainful employment in all. The input is a set of 4 (positive!) rates
*R*_{k} for the 4 categories of crime.
- Include that subroutine in a larger one that
- creates and follows —say— a population of 100 criminals, and
- computes population averages of their total income and total time spent in jail.

- Thus the average income <
*I*> and average (total) time in jail <*J*_{tot}> are functions of the vector *R*. You can now use a multivariable minimizer to maximize the ratio <*I*>/<*J*_{tot}> .