This course is an introduction to discrete probability focusing on Bernoulli trials and associated distributions. It introduces basic random processes such as the Poisson process, random walk, and Markov chains. The simulation of random variables and random processes is emphasized throughout the course using the software R.

Prerequisites

In addition to the common requirements, students should have taken an elementary probability or probability and statistics course.

Some elementary knowledge of R is also required, but this will be covered during the optional preliminary workshop on computational tools.

On completion of the course, students will be able to:

• arrow_rightunderstand and use the concepts of probability space, random variable, probability distribution, expected value, independence and random process and will have been exposed to concrete examples of these concepts;
• arrow_righthandle the main simulation procedures for random variables and use the methods implemented in R to this effect;
• arrow_rightunderstand the proof of two basic theorems of probability theory: the law of large numbers and the central limit theorem, in the context of independent Bernoulli trials, and be familiar with some of their applications;
• arrow_rightdefine and use random processes such as the Poisson process and discrete Markov chains, as well as use them as models for concrete problems and simulate them to study their properties; 
• arrow_rightunderstand processes and random variables to model, simulate, and analyze practical problems.

1. Review of Basic Probability Concepts (3 weeks)

Probability spaces. Conditional probability and independence. Random variables, distribution functions, discrete and continuous variables, expected value, moments, Markov and Chebyshev inequalities. Probability generating functions.

2. Random Variable Simulation (1 1/2 weeks)

Basic concepts about random number generators. The Inverse Transform method with examples. The acceptance-rejection method with examples. Particular methods: random variables with normal, Poisson, or binomial distribution. Simulation of random variables with R.

3. Bernoulli Trials (3 weeks)

Definition and related distributions: Binomial, geometric, negative binomial, Poisson. The Law of Large Numbers, Chernoff bounds, Stirling’s formula, the Central Limit Theorem (DeMoivre-Laplace) and applications. Simulation of Bernoulli trials and related variables.

4. Poisson Processes (2 weeks)

Exponential distribution. Poisson process, definition, and characterizations. Distributions related to a Poisson process. Compound Poisson processes. Decomposition and superposition of Poisson processes. Non-homogeneous processes. Simulation and applications.

5. Discrete Markov Chains (3 weeks)

Definition and examples. Transition matrices, Chapman-Kolmogorov equations, classification of states. Stationary distributions. Example: Simple random walk, properties, simulation, Arcsine Law. Simulation of Markov Chains with applications.

6. Discrete Event Simulation (1 1/2 weeks)

Introduction to the discrete event simulation method. Queue with one server, queue with two serial or two parallel servers, simple inventory models and the insurance risk model. Statistical analysis of simulated data.

Bibliography

1. Baclawski, Kenneth (2008). Introduction to Probability with R, Boca Raton: Chapman & Hall/CRC.

2. Dekking, F.M.; Kraaikamp, C.; Lopuhaä, H.P.; Meester, L.E. (2010). A Modern Introduction to Probability and Statistics, London: Springer.

3. Dobrow, Robert P. (2014). Probability with Applications and R, Hoboken: Wiley.

4. Dobrow, Robert P. (2016). Introduction to Stochastic Processes with R, Hoboken: Wiley.

5. Durret, Rick (2009). Elementary Probability for Applications Cambridge University Press.

6. Jones, Owen; Maillardet, Robert; Robinson, Andrew (2009). Introduction to Scientific Programming and Simulation Using R, Boca Raton: CRC Press.

7. Karlin, Samuel; Taylor, Howard M. (1998) An Introduction to Stochastic Modeling, 3rd. Edition, Academic Press.

8. Lesigne, Emmanuel (2005). Head or Tails: An Introduction to Limit Theorems in Probability, American Mathematical Society.

9. Ross, Sheldon M. (2012). Simulation, 5th. Edition, Academic Press.

10. Suess, Eric A.; Trumbo, Bruce E. (2010). Introduction to Probability, Simulation and Gibbs Sampling with R, New York: Springer.

Books on R

1. Adler, Joseph (2010). R in a Nutshell, Beijing: O’Reilly.

2. Matloff, Norman (2011). The Art of R Programming, San Francisco: No Starch Press.

3. Murrell, Paul (2011). R Graphics, 2nd. Edition,  Boca Raton: CRC Press.

4. Zuur, Alain; Leno, Elena N.; Meesters, Erik (2009). A Beginner’s Guide to R, Dordrecht: Springer.

Support Sessions

2 hours per week with a teaching assistant.

Grading

Two midterm exams (40%), homework (40%), a simulation project to be submitted/ presented by the end of the course (20%).