Andrei Antonenko

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Probability Theory

This is a full set of lecture notes and problem sets for a Probability Theory class (AMS 311) taught by me in Spring 2005 at the Department of Applied Mathematics and Statistics at Stony Brook University. There probably are some typos. Feel free to contact me any time if you see any, or just with your comments/suggestions!

Lecture Notes

Lecture 01Introduction. Set Theory. Probabilistic Models.
Lecture 02Probabilistic Models: Examples. Probability Law. Conditional Probability.
Lecture 03Multiplication Rule. Total Probability Theorem. Bayes’ Rule.
Lecture 04Independence.
Lecture 05Conditional Independence. Counting. Bernoulli Trials.
Lecture 06Counting: Partitions. Discreet Random Variables. Probability Mass Function.
Lecture 07Functions of Random Variables. Expectation and Variance.
Lecture 08Expectation and Variance. Joint PMF.
Lecture 09Conditioning. Conditional Expectation.
Lecture 10-11Conditional Expectation. Independence of Random Variables.
Lecture 12General Random Variables. Probability Density Function. Expectation. Exponential Random Variable. Cumulative Density.
Lecture 13Distributions of Maximum and Minimum. Normal Random Variable. Conditioning on Event.
Lecture 14Conditional Expectation.
Lecture 15Multiple Random Variables. Conditioning. Expectation. Continuous Bayes’ Rule.
Lecture 16Independence. Joint CDF. Derived Distributions.
Lecture 17Convolutions.
Lecture 18Transforms. Moments. Inversion Property.
Lecture 19Transform of the Sum. Conditional Expectation as a Random Variable.
Lecture 20Conditional Variance as a Random Variable. Sum of Random Number of Random Variables.
Lecture 21Sum of Random Number of Random Variables. Covariance.
Lecture 22Covariance and Correlation. Estimation.
Lecture 23-24Stochastic Processes. Bernoulli Process.
Lecture 25Poisson Processes.
Lecture 26Poisson Processes: Time of the First Arrival, Interarrival Times.
Lecture 27Merging Processes. Competing Exponents. Incidence Paradox.
Lecture 28Markov Chains.
Lecture 29Markov Chains: Classification of States, Periodicity, Steady-State Behavior.
Lecture 30Birth-Death Processes.
Lecture 31Absorption Probabilities.
Lecture 32Expected Times to Absorption. Mean First Passage Time.
Lecture 33Limit Theorems. Inequalities: Markov, Chebyshev.
Lecture 34Weak Law of Large Numbers.
Lecture 35Central Limit Theorem.
Lecture 36Polling.

Problem Sets

Problem Set 01Set Theory. Probability Axioms. Basics of Conditional Probability.
Problem Set 02Conditional Probability. Bayes’ Rule. Independence.
Problem Set 03Counting. Discrete Random Variables. PMFs. Expectations.
Problem Set 04Expectations. Variance. Joint PMFs. Conditioning. Independence.
Problem Set 05Continuous Random Variables. PDFs. Expectation. Variance.
Problem Set 06Continuous Random Variables. Derived Distributions. Conditional Expectation and Variance. Transforms.
Problem Set 07Bernoulli Processes. Poisson Processes.
Problem Set 08Markov Chains.
Problem Set 09Limit Theorems.

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About me

Andrei Antonenko

Lecturer, Department of Linguistics
Stony Brook University

Ph.D. 2012, Stony Brook University

Areas of interest: syntax, semantics, linguistics education

 

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