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 01 | Introduction. Set Theory. Probabilistic Models. |

Lecture 02 | Probabilistic Models: Examples. Probability Law. Conditional Probability. |

Lecture 03 | Multiplication Rule. Total Probability Theorem. Bayes’ Rule. |

Lecture 04 | Independence. |

Lecture 05 | Conditional Independence. Counting. Bernoulli Trials. |

Lecture 06 | Counting: Partitions. Discreet Random Variables. Probability Mass Function. |

Lecture 07 | Functions of Random Variables. Expectation and Variance. |

Lecture 08 | Expectation and Variance. Joint PMF. |

Lecture 09 | Conditioning. Conditional Expectation. |

Lecture 10-11 | Conditional Expectation. Independence of Random Variables. |

Lecture 12 | General Random Variables. Probability Density Function. Expectation. Exponential Random Variable. Cumulative Density. |

Lecture 13 | Distributions of Maximum and Minimum. Normal Random Variable. Conditioning on Event. |

Lecture 14 | Conditional Expectation. |

Lecture 15 | Multiple Random Variables. Conditioning. Expectation. Continuous Bayes’ Rule. |

Lecture 16 | Independence. Joint CDF. Derived Distributions. |

Lecture 17 | Convolutions. |

Lecture 18 | Transforms. Moments. Inversion Property. |

Lecture 19 | Transform of the Sum. Conditional Expectation as a Random Variable. |

Lecture 20 | Conditional Variance as a Random Variable. Sum of Random Number of Random Variables. |

Lecture 21 | Sum of Random Number of Random Variables. Covariance. |

Lecture 22 | Covariance and Correlation. Estimation. |

Lecture 23-24 | Stochastic Processes. Bernoulli Process. |

Lecture 25 | Poisson Processes. |

Lecture 26 | Poisson Processes: Time of the First Arrival, Interarrival Times. |

Lecture 27 | Merging Processes. Competing Exponents. Incidence Paradox. |

Lecture 28 | Markov Chains. |

Lecture 29 | Markov Chains: Classification of States, Periodicity, Steady-State Behavior. |

Lecture 30 | Birth-Death Processes. |

Lecture 31 | Absorption Probabilities. |

Lecture 32 | Expected Times to Absorption. Mean First Passage Time. |

Lecture 33 | Limit Theorems. Inequalities: Markov, Chebyshev. |

Lecture 34 | Weak Law of Large Numbers. |

Lecture 35 | Central Limit Theorem. |

Lecture 36 | Polling. |

*Problem Sets*

Problem Set 01 | Set Theory. Probability Axioms. Basics of Conditional Probability. |

Problem Set 02 | Conditional Probability. Bayes’ Rule. Independence. |

Problem Set 03 | Counting. Discrete Random Variables. PMFs. Expectations. |

Problem Set 04 | Expectations. Variance. Joint PMFs. Conditioning. Independence. |

Problem Set 05 | Continuous Random Variables. PDFs. Expectation. Variance. |

Problem Set 06 | Continuous Random Variables. Derived Distributions. Conditional Expectation and Variance. Transforms. |

Problem Set 07 | Bernoulli Processes. Poisson Processes. |

Problem Set 08 | Markov Chains. |

Problem Set 09 | Limit Theorems. |