MA540 Introduction to Probability Theory
Instructor: Nikolay S. Strigul
Office Hours: by appointment
Lectures: Pierce 116, Monday 6:15 pm -8:45 pm
Assignments, Exams, Quizzes: There will be regular homework assignments, midterm and final exams.
Every class will start with a closed book quiz consisting of definitions and theorems.
- Quizzes: 20 %
- Homework assignments: 20 %
- Midterm: 20 %
- Final: 40 %
MA540 is a graduate-level introduction in probability theory. Compared to undergraduate-level
courses in probability and statistics, all theorems will be rigorously proven—however we will
not use the measure theory as in more advanced courses. Topics will include sample space and events,
combinatorial methods for computing probabilities, random variables, expectations, limit theorems,
and stochastic processes. Theoretical problems will be illustrated by computer simulations using
Mathematica software. We will follow William Feller's classic "An Introduction to Probability
Theory and Its Applications". Homework assignments will also have both theoretical and computational
components. For the computational part we will use Jenny A. Baglivo's "Mathematica Laboratories
for Mathematical Statistics: Emphasizing Simulation and Computer Intensive Methods".
1) An Introduction to Probability Theory and Its Applications
by William Feller, V. 1, ISBN: 0471257087
2) Mathematica Laboratories for Mathematical Statistics:
Emphasizing Simulation and Computer Intensive Methods by Jenny A. Baglivo, ISBN: 0898715660
Lecture 1. - The Nature of Probability Theory.
1. "Statistical probability"
2. The empirical background.
3. The sample space, events.
4. Relations among events.
5. Introduction to Mathematica.
Lecture 2. - Elements of Combinatorial Analysis.
6. Subpopulations and partitions.
7. The hypergeometric distribution.
8. Binomial coefficients.
9. Stirling's formula.
Lecture 3. - Fluctuations of Coin Tossing and Random Walks.
10. The reflection principle.
11. Random walks.
12. Computer simulations of random walks.
Lecture 4. - Combination of events. Stochastic independence.
13. Union of events.
14. The realization of m among N events.
15. Conditional probability.
16. Stochastic independence.
17. Application to genetics.
Sep 26. Lecture 5. - The Binomial distribution.
18. Bernoulli trials.
19. The Binomial distribution.
20. The central term and the tails.
21. The Law of Large Numbers.
Lecture 6. - The Poisson distribution.
22. The Poisson approximation to the Binomial distrinution.
23. The Poisson distribution.
24. Applications of the Poisson distribution.
25. The Negative Binomial distribution.
Lecture 7. - The Normal Approximation to the Binomial Distribution.
26. The normal distribution.
27. The DeMoivre-Laplace Limit Theorem.
28. Relation to the Poisson Approximation.
Lecture 8. - Random Variables.
29. Random variables.
33. Chebyshev's Inequality.
Lecture 9. - Laws of Large Numbers.
34. Identically distributed variables.
35. Proof of the law of large numbers.
36. The theory of 'fair' games.
37. Variable distributions.
38. The strong law of large numbers.
Lecture 10. - Generating functions.
40. Waiting times in Bernoulli trials.
41. Partial fraction expansions.
42. Bivariate generating functions.
Lecture 11. - Random walks.
43. The classical ruin problem.
44. Application of generating functions.
45. Connections with diffusion process.
46. Random walks in plane and space.
Lecture 12. - Markov Chains I.
47. Definitions and illustrative examples.
48. Transition probabilities.
49. Classification of states.
50. Irreducible chains. Decomposition.
Lecture 13. - Markov Chains II.
51. Transient chains.
52. Periodic chains.
53. The general Markov process.
Lecture 14. - Time-dependent stochastic processes.
54. The Chapman-Kolmogorov identity.
55. The Poisson process.
56. The backward equations.