Some Topics in Probability

Basic Facts

• Introduction to probability

Independence

• Independence of Random Variables
• Pairwise and 3-way independence for events

Conditional Probabilities and Bayes Rule

• Somewhat unintuitive consequences for games
• Examples of applicatipns of conditional probabilities and Bayes Rule

The Random Walk

• Random Walks and Difference Equations
• Gambler's Ruin
• Gambler's Ruin Part 2: hitting times

Limit Theorems

• Review of facts about expectation, limit theorems and more
• Continuous random variables as limits
• The Central Limit Theorem
• The law of Large Numbers and the Central Limit Theorem
• A "naive" observation on the Gaussian distribution

Miscellaneous

• Comments on complete additivity of probabilities
• Complements: Inclusion-Exclusion; The voting problem
• The Hypergeometric Distribution
• Discrete Random Variables
• More facts about distributions. The Poisson distribution
• More facts about discrete distributions
• Continuous Random Variables
• More facts about continuous random variables
• Application to genetics (Mendel's Law)
• An application to queuing theory
• Short comments on important topics that complement a first course in probability