Longhai Li, Department of Mathematics and Statistics, University of Saskatchewan
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.
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.
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