Introduction to Probability Theory

**Thursday 9/27:** Review of the course information and admin details. Introduction and basic concepts: probability as a measure of uncertainty; interpretations of probability; experiments, samples spaces, and events. Reading: Sections 1.1-1.3 (lecture notes)

**Tuesday 10/2:** Examples for experiments, sample spaces and events. Definitions and operations from set theory. Axiomatic definition of probability. Properties of probability. Reading: Sections 1.3-1.5 (lecture notes)

**Thursday 10/4:** Properties of probability: probability of the union of two events and of three events; bounds for the probability of a general finite union or finite interesection (Bonferroni inequality). Finite sample spaces; computing probabilities in simple sample spaces through counting. Reading: Sections 1.5-1.6, Section 1.10 (only for the union of three events) (lecture notes)

**Tuesday 10/9:** Counting methods: multiplication rule; permutations; combinations (binomial coefficient); extension of combinations to more than two groups (multinomial coefficient). Reading: Sections 1.7, 1.8, 1.9 (lecture notes)

**Thursday 10/11:** Conditional probability: motivation, definition, and examples. Multiplication rule for conditional probabilities. Law of total probability. Bayes theorem. Reading: Sections 2.1, 2.3 (lecture notes)

**Tuesday 10/16:** Independent events: definitions through conditional probabilities and through probabilities of intersections, properties, and examples. Conditional independence. Reading: Section 2.2 (lecture notes)

**Thursday 10/18:** Bayes theorem illustration through the diagnostic test example. Binomial distribution. Statistical inference for the Binomial probability of success, using Bayes theorem. Reading: Sections 2.1, 2.2, 2.3 (lecture notes)

NOTE:** ***As we develop the methods of chapter 3, we will be working with several special distributions (both discrete and continuous) that are discussed in more detail in chapter 5. For reading, there will therefore be references to sections of chapter 5. The material in chapter 5 includes additional results on expectations and variances, which are concepts that will be studied in chapter 4. When we get to chapter 4, we will discuss the corresponding results. Therefore, the course material from chapter 5 will be developed in the context of the general methods to be covered from chapters 3 and 4.*

**Tuesday 10/23:** Random variables and distributions. Discrete and continuous random variables; probability functions and probability density functions. Examples of discrete distributions. Geometric distribution. Reading: Sections 3.1, 3.2, 5.2 (lecture notes)

**Thursday 10/25:** Continuous random variables and probability density functions. Examples of continuous distributions: uniform, exponential, and normal distributions. Definition and properties of the cumulative distribution function. Reading: Sections 3.2, 3.3, 5.6, 5.7 (lecture notes)

**Tuesday 10/30:** Exam 1

**Thursday 11/1:** Cumulative distribution functions: properties and examples for discrete and continuous random variables. Obtaining the probability function or the probability density function from the cumulative distribution function. Quantile functions and percentiles. Reading: Section 3.3 (lecture notes)

**Tuesday 11/6:** Bivariate distributions. Joint p.f. for discrete random variables, joint p.d.f. for continuous random variables, and joint p.f./p.d.f. for mixed discrete/continuous random variables. Bivariate cumulative distribution functions. Reading: Section 3.4 (lecture notes)

**Thursday 11/8:** Marginal distributions for discrete, continuous and mixed discrete/continuous bivariate random variables. Independent random variables. Definitions for conditional distributions. Reading: Sections 3.5, 3.6 (lecture notes)

**Tuesday 11/13:** Conditional distributions, multiplication rule for distributions, Bayes theorem for random variables. Bayesian inference for the probability of success of a Binomial distribution, and for the rate parameter of the Poisson distribution (Beta and gamma distributions). Reading: Sections 3.6, 5.4, 5.7, 5.8 (lecture notes)

Chapter 3 summary (Sections 3.1 -- 3.6)

Definitions and formulas from Section 3.7 (this is the only material from Section 3.7 that will be included in the exams; we mainly need the definition of independence and of a "random sample" for a finite collection of random variables)

**Thursday 11/15:** Basic definitions for multivariate distributions (refer to the handout above). Functions of a random variable: transformation methods for discrete and continuous random variables, techniques and examples. Reading: Section 3.8 (lecture notes)

**Tuesday 11/20:** Random variate generation for continuous distributions, using the probability integral transformation. Distribution of the sum of two independent continuous r.v.s (convolution); distribution of the maximum and the minimum of a random sample from a continuous distribution. Reading: Sections 3.8, 3.9 (for section 3.9, focus on convolutions, and the distribution of the maximum and minimum) (lecture notes)

**Thursday 11/22:** *Thanksgiving break*

Tuesday 11/27: Exam 2

**Thursday 11/29:** The expectation as a measure of center for the distribution of a random variable. Definitions, interpretation, and examples. Expectations for functions of random variables. Properties of expectations. Reading: Sections 4.1, 4.2 (lecture notes)

**Tuesday 12/4:** The variance as a measure of spread for the distribution of a random variable. Definition and properties. The median and the interquartile range as alternative measures of center and dispersion. Expectation and variance for the Binomial, Poisson, uniform, normal, and gamma distributions. Reading: Sections 4.1 -- 4.3 (and 5.2, 5.6, 5.7) (lecture notes)

**Thursday 12/6:** Probabilistic analysis of profit from games of chance (case study with roulette bets), using expectations, standard deviations, and the Central Limit Theorem. Optional reading: Sections 6.2, 6.3 (lecture notes)