STAT 342 Mathematical Statistics

Description

This course deals with basic probability concepts at a moderately rigorous level. Topics include: probability spaces, conditional probability and independence, discrete and continuous random variables, standard probability models, expectations, moment generating functions, transformation of random variables, sampling distributions, limiting theory, and elementary statistical inference. The lecture notes follows closely to the textbook: Introduction to Mathematical Statistics by Hoggs, Mckean, and Craig.

List of Lectures and Topics

Introduction to Statistical Inference

Lecture 1: Introduction

Probability Theory for A Single Random Variable

Lecture 2: Axioms of Probability

Lecture 3: Conditional Probability and Bayes Rule

Lecture 4: Independence of Events

Lecture 5: Random Variables and Cumulative Distribution Function (I)

Lecture 6: Random Variables and Cumulative Distribution Function (II)

Lecture 7: Probability Density Function and Transformation of a Single Random Variable

Lecture 8: Expectation, Moment Generating Function, Chebyshev Inequalities

Lecture 9: Jensen Inequality and Joint Distribution

Random Vector

Lecture 10-11: Expectation for Joint Distribution, Moment Generating Function, and Transformation of Random Vector

Lecture 12: Conditional Expectation, Iterative Expectation Formula, Decomposition of Variance

Lecture 13: Correlation coefficient and Independence of Random Variables

Lecture 14: Extensions to More than 2 Random Variables, Linear Combination

Lecture 15: Bernoulli, Binomial, Multinomial Distribution

Lecture 16: Poisson and Gamma

Lecture 17: Exponential, Erlang, Poisson Process, Normal Distribution

Lecture 18: Chi-square distribution, Student's Theorem

Lecture 19: t and F distribution, Mixture distribution

Asymptotic Theory

Lecture 20: Convergence in Probability, Law of Large Numbers, Consistency of sample mean and Variance

Lecture 21: Convergence in Distribution, Central Limit Theorem, Continuous Mapping and Slustky Theorems, Applications to Sample Means.