**Lectures:** Wednesday, 4:30-7:10pm, Research I, room 302

This course covers applications of Monte Carlo methods in science. The course begins with a review of relevant background material in probability and statistics, and then covers algorithms for computer generation of random numbers, first from a uniform distribution, and then from various distributions of interest. Examples of applications from the physical, biological, and social sciences will be studied. Individual class projects will focus on topics chosen by the students.

Gentle, James E. (2003), Random Number Generation and Monte Carlo Methods, second edition, Springer, New York.

This will be supplemented by various handouts.

- Some basics of probability
- Random variables and their relationships to each other
- Some basics of statistical estimation
- Generating random numbers on the computer
- First simple examples: integration by Monte Carlo
- Principles and examples of mathematical modeling
- Discrete simulation models
- Stochastic diffusion models
- Markov chain Monte Carlo
- Stochastic optimization

Background and interests of class will affect the relative weighting of these topics.

www.r-project.org

From that web page, go to CRAN, select a mirror, then select the appropriate distribution. I will introduce the system in the first class.

Assignments will require the use of computer software that has facilities for generating random numbers and for general programming. Instead of R, this could be Matlab, Mathematica, Maple, C, and/or Fortran.

The GMU Honor Code will be strictly observed.

- Some basics of probability theory
- Distributions of transformed random variables
- The PDF decomposition

Notes on probability distributions

**Assignment 1**

Due January 28.

- Statistical estimation Notes
- Monte Carlo estimation
- Computer generation of uniform random numbers
- Algorithms
- Software slides on R from lecture not covered in class.

**Assignment 2:** Page 271: Exercises 7.1, 7.2, 7.3, 7.5;
Page 57: Exercises 1.4, 1.5.

Due February 11.

- Monte Carlo methods.
- Monte Carlo studies in statistics.
- Computer generation of uniform random numbers. Software for random number generation. (Also see Chapter 8.)
- R intro.
- More on R.
- Term project.

- Term project
- More on computer generation of uniform random numbers
- Generation of nonuniform random numbers

**Assignment 3:** Page 159: Exercises 4.1, 4.2, 4.3, 4.4, 4.6, 4.7, 4.9 (Instead of Fortran
or C, you can use R or some other package.)

Due February 25.

Comments, solutions. Also, inverse CDF for bivariate
distribution.

- Term project (brief written plan)
- Generation of nonuniform random numbers

Slides from lecture (to be continued next week).

- Generation of nonuniform random numbers

This will cover basic material on probability distributions, Section 7.1, Chapter 1 (but not Exercise 1.4), and Sections 4.1-4.7.

The exam questions will be similar to the (easier) homework exercises, except you will not be asked to use any computer software.

- Review exam
- MCMC (Metropolis, Metropolis-Hastings, Gibbs, hit-and-run)

Slides from lecture.

**Assignment 4:** Page 161: Exercises 4.10, 4.11, 4.15

Due April 8. **Notice: a lot of homework is due on that day!**

- More on general methods of random number generation
- Applications in Bayesian statistical methods

Slides from lecture.

**Assignment 5:** Page 162: Exercises 4.12, 4.14, 4.17

Due April 8. **Notice: a lot of homework is due on that day!**

- Applications in Bayesian statistical methods
- MCMC

Slides from lecture.

Demo R program.

- Tests of random number generators

Slides from lecture. - Simulation models

**Assignment 6:** Exercises 2.1, 2.8, and 3.6

Due April 12.

- Quasirandom numbers.

Slides from lecture. - Simulation models

- Discuss homework

code for 2.1 - Term project presentations

- Term project
- Simulation models

James Gentle, jgentle at gmu dot edu