Advanced Probability Theory

Overview

The course introduces the concept of probability through Kolmogorov’s Axioms. It develops the concept of probability density function, cumulative distribution function, and introduces the concept of a random variable. Different theoretical probability distributions, both discrete and continuous are introduced, and their properties are studied. Different generating functions, viz. MGF, PGF, and characteristic functions of different variables are discussed. It also introduces functions of one or two random variables, and derived random variables, such as t, chisq, F are studied in details. The concept of Order Statistics is introduced, and derivation of distribution of range, median etc. are discussed. Different limit theorems are discussed in detail.INTENDED AUDIENCE : Undergraduate and postgraduate students in statistics, mathematics, and Machine learningPREREQUISITES : Basics of Real Analysis, Functions of two variables, Convergence of a function.INDUSTRY SUPPORT : Most Financial companies

Syllabus

Week 1 :Introduction, Sample Space, Probability Axioms, Theorems on Union and Intersections of events in a Sample Spaces. Bertrand’s Paradox.Week 2 :Conditional Probability, Bayes Theorem, Probability on Finite Sample Spaces. Independence of Events..Week 3 :Introduction to Random variables – discrete & continuous Random variables Discrete random variables - Uniform, Bernoulli, Binomial, Geometric, Poisson Distributions, Hypergeometric, Negative Binomial
Week 4 :Continuous Random variables: Uniform, Normal, Exponential, Gamma, Cauchy, Beta1 and Beta2Week 5 :Moments of a distribution, Bivariate distribution, Covarience and Correlation
Week 6 :Generating Functions and their properties: Moment Generating Function Characteristic Functions and Probability Generating Function
Week 7:Poisson Process, Conditional Expectations and Variance, Chebyshev's Inequalityand Introduction to Bivariate Normal.Week 8 :Functions of Random Variables, Introductionto t and F distribution.
Week 9 :Order Statistics
Week 10 :Limit Theorems: Mode of Convergence
Week 11 : Laws of Large numbers
Week 12 :Central Limit Theorems