Probability Theory I

Syllabus

• Probability basics; Kolmogorov axioms; combinatorial probability; geometric probability models.
• Conditional probability; Bayes’ theorem; law of total probability; independence.
• Conditional expectation (on positive-probability events) and the law of total expectation; simple forecasting ideas.
• Random walks and ruin probabilities (introductory results).
• Random variables (and vectors): distribution and density functions; independence; distribution of sums of independent variables.
• Classical discrete and continuous distributions (overview).
• Generating functions (intro).
• Expectation and variance: properties and computation; classical inequalities; median and moments; covariance and correlation.
• Law of Large Numbers (weak and strong); Central Limit Theorem (statement).