Discrete Probability and Simulation
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.
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:
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 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.
Books on R
2 hours per week with a teaching assistant.
Two midterm exams (40%), homework (40%), a simulation project to be submitted/ presented by the end of the course (20%).