Last changed: Sep 10, 2000, 07:43 PM

Topic 1 . Topic 2 . Topic 3 . Topic 4 . Topic 5 . Topic 6 . Topic 7 . Topic 8 . Topic 9 .


    Required: Ross, S., "A First course in Probability" Fifth edition, 1998, Prentice Hall

    Recommended: Feller, W, "An Introduction to Theory of Probability", vol. I 3rd edition, 1971, John Wiley

This course is required for Ph.D. students in MSIS department. We will have a final exam and several home work assignments. The exam will constitute 40% of your grade and the home works 50%. The remaining 10% will be used for scribing duties.

The lectures will cover the main topics in the textbook plus additional topics not covered in there. For these latter lectures I will ask one of the students to take careful notes, and transcribe them and then have them ready for posting in the notes page of the web page. Scribing duties will constitute 10% of your grade.

Here are a tentative list of topics and outline of a syllabus:

Topic 1: (1 lecture)

    Elements of probability theory, Bernoulli trials

    Principle of inclusion/exclusion

    Discrete random variable; probability mass and distribution functions; expectation, moments, variance, and other cumulants

    Binomial, Poisson, geometric, negative binomial and hypergeometric random variables

Topic 2: (1 lecture)

    Ordinary generating functions (OGF), calculation of expectation, variance, moments and cumulants with OGF

    Functions of random variables, sums of random variables, techniques for using OGFs

Topic 3: (1 lecture)

    Markov, chebychev inequalities and Chernoff bounds

    Conditional random variables; independence, Bay's rule and applications


Topic 4: (2 lectures)

    Markov chains: basic definitions, matrix representation, significance of eigenvalues and eigenvectors

    Generating functions and Markov chains

    Reversible Markov chains and random walks

Topic 5: (1 lecture)

    Simulation of random variables, pseudo random number generators

    Random permutations, random subsets (and subsets of a prescribed size), random partitions

Topic 6: (2 lectures)

    Continuous random variables

    Notion of measure, Borel sets and Lebesgue integrals

    Stieltjes integrals and unification of discrete and continuous notations

    continuous random variables, probability density and distribution functions

Topic 7: (2 lectures)

    OGF, moment generating and characteristic functions

    Expectation, variance, moments and cumulants and connection to generating functions

    Exponential random variables, waiting times, gamma random variables

    Normal, and exponential random variables

    Functions of random variables

    Normal random variables

    Beta, chi-square, t, F and other statistical random variables

Topic 8: (1 lecture)

    Conditional distributions, independence, joint distributions

    Examples from joint normal and joint exponential distributions

    Gamma, beta, chi-square, t, F and other statistical random variables

Topic 9: (1 lecture)

    Conditional distributions, independence

    Simulation of continuous random variables, in particular normal distribution

Topic 10: (2 lectures)

    Central limit theorem

    Law of large numbers (weak and Strong versions)

    Law of iterated logarithm

    Other limit theorems