Probability theory is the branch of mathematics dealing with modeling uncertainty. It is important because of its direct applications in areas such as genetics, finance, and telecommunications. It also forms the basis of many other areas in the mathematical sciences, including statistics, modern optimization methods, and risk modeling. This course introduces probability theory, random variables, and Markov processes. Topics covered include: Axioms of Probability, Conditional Probability; Bayes’ Theorem; Discrete Random Variables, Moments, Boundary Probability, Probability Generating Functions, Standard Discrete Distributions; Continuous Random Variables, Uniform, Normal, Cauchy, Exponential Distributions , gamma and chi-square distributions, transformations, Poisson processes; bivariate distributions, marginal and conditional distributions, independence, covariance and correlation, linear combinations of two random variables, bivariate normal distributions; autorandom variables Sequences, weak laws of large numbers, central limit theorem; definitions and properties of Markov chains and probability transition matrices; solving equilibrium equations.
1 Basic probability axioms and rules and moments of discrete and continuous random variables, and familiarity with commonly named discrete and continuous random variables.
2 How to derive probability density functions for transformations of random variables and use these techniques to generate data from various distributions.
3 How to calculate probabilities and derive marginal and conditional distributions for binary random variables.
4 Discrete-time Markov chains and methods for finding equilibrium probability distributions.
5 How to calculate the absorption probability and expected hit time for a discrete-time Markov chain with absorbing states.
6 How to translate real-world problems into probabilistic models.
7 How to read and annotate an outline of a proof and be able to write a logical proof of a statemen
