Discrete Probability and Simulation
COURSE DESCRIPTION 
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 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. 
COURSE GOALS 
On completion of the course, students will be able to:

COURSE CONTENT 
Probability spaces. Conditional probability and independence. Random variables, distribution function, 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 acceptancerejection 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 (DeMoivreLaplace) 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. Nonhomogeneous processes. Simulation and applications. 5. Discrete Markov Chains (3 weeks) Definition and examples. Transition matrices, ChapmanKolmogorov 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
Books on R
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%). 