## Introduction to Probability

Date Main lecture topics Relevant reading Homework due
(posted on Collab)
1/14 Hello. Syllabus. Questionnaire. Mathematical setup for a probability experiment: sample space, events, probability measure and its properties. Examples. Cartesian products. Sampling with and without replacement. Classwork. 1.1-2
1/16 Review of last lecture and some parts of classwork. How to count some things. Some infinite sample spaces: flipping until a head, and choosing a point uniformly at random. Clickers. 1.2-3
1/21 Counting and decision trees. Why points have zero probability when choosing uniformly at random from an interval or 2D region. Probabilities and parities. Some rules and Venn diagrams. Clickers throughout. 1.3-4
1/23 Inclusion-exclusion. Law of Large Numbers as the meaning of the probability of an event. Basic terminology of random variables, examples, and classwork. 1.5 HW 1
1/28 Review of some classwork from last time: inclusion-exclusion and a pmf. Conditional probability: definition, multiplicative rule, decision trees and the law of total probability. Some practice problems, leading to Bayes' formula. 2.1-2
1/30 Review of previous lecture, with practice. Bayes' formula. Independence for events and r.v. Bernoulli and binomial r.v. 2.2-4 HW 2
2/4 Geometric r.v. Probability density functions, properties and examples, including the uniform distribution. The Monty Hall problem. Clickers and classwork. 2.4, 3.1
2/6 Clickers: the birthday problem, independence versus pairwise independence. Review of density functions. Cumulative distribution functions. Classwork. 2.5: conditional independence and the birthday problem, 3.2 HW 3
2/11 Conditional independence. Expectations of discrete and continuous r.v., also expectations of functions of such. Expectations don't always exist. Computing cdfs and extracting pmfs, pdfs, and probabilities from them. Classwork. 3.2-3
2/13 Examples: distance to the center of a dartboard, amount paid in the presence of a deductible, expectation of X ~ Bin(n,p). Classwork. 3.2-3 HW 4
2/18 Another example: length of the longer piece of a broken stick. Terminology: moments, median, quantiles, variance, standard deviation. Revisiting a HW problem. Classwork. 3.2-4
2/19 Midterm 1, 7-8:30 PM, Olsson 120
2/20 Review of 3.2-3. Facts about variance. The standard normal: density, cdf, and properties. Other normal r.v. The Central Limit Theorem for binomial r.v. Classwork. 3.4-5,4.1
2/25 Review of variance, standard normal, other normals, CLT. Normal approximation to the binomial, with examples and classwork. 3.5,4.1
2/27 Law of Large Numbers for binomial r.v. Percentage of the sample within one or two standard deviations of the mean, for a normal r.v. Review of the normal approximation to the binomial, with a condition to check, the continuity correction, and a variation for random walks. Clickers. 4.1-2 HW 5
3/3 LLN: meaning, proof. Confidence intervals. Poisson r.v. and the Law of Rare Events. Clickers. 4.2-4
3/5 A HW problem. Proof of LRE. Using Poisson to estimate the number of occurrences of rare approximately independent events. Exponential r.v. and their use as approximate waiting times. Clickers. 4.4-5 HW 6
The course moved online during spring break. Course information is now kept exclusively on Collab.

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