Math 550 syllabus
The purpose of this course is to give students an introduction to probability theory and probability distributions. In particular, we will explore the axiomatic approach to probability, counting techniques, Bayes Theorem, random variables, probability distributions for discrete and continuous random variables, mathematical expectation, moment generating functions, joint and conditional distributions for multiple random variables, and measures of association (covariance and correlation), the law of large numbers and the central limit theorem.
A comprehensive introduction to probability, the mathematical theory used to model uncertainty, covering the axioms of probability, random variables, expectation, classical discrete and continuous families of probability models, the law of large numbers and the central limit theorem.
Prerequisites: MA 227 and C or better in MA 237. Credit for both MA 550 and MA 451 is not allowed.
Textbook: Mathematical Statistics with Applications, 7th edition by Dennis D Wackerly, William Mendenhall, Richard L Scheaffer. Duxbury Press. ISBN #0495110817
Course Coverage: Chapters 1-7.
Calculator Prerequisite A scientific calculator is required; the TI-89 is recommended. Project Examples
Absorbing Markov chains. Markov Chains. A discrete (discrete-time) random process with the Markov property. Transition matrix. Examples of Markov chains. Absorbing Markov chains. Fundamental Matrix. Time to absorption. Absorption probabilities.
Ergodic Markov chains. Markov Chains. A discrete (discrete-time) random process with the Markov property. Transition matrix. Examples of Markov chains. Ergodic Markov chains. Regular Markov chains. Law of Large Numbers for Ergodic Markov chains. Fundamental limit theorem for regular chains.
Random Walks. Returns and First Returns. Probability of Eventual Return. Expected Number of Equalizations. Gambler's Ruin. Arc Sine Laws.
Generating Functions for Discrete Distributions. Generating Functions for Discrete Distributions. Moment Generating Functions. Ordinary Generating Functions.
Generating Functions for Continuous Densities. Generating Functions for Continuous Densities. Moment Generating Functions. Moment Problem. Cauchy Density.
Branching Processes. Generating Functions for Discrete Distributions. Branching Processes. Problem of Extinction. Distribution of Offspring. Chain Letter Problem.
Random Matrices. Wigner matrices and Moments Estimation. Wigner's theorem. Free Probability. Freeness. Free probability. The combinatorics of freeness.