Probability Theory

Probability is the mathematics of random events. In the context of modern statistics, probability theory provides a framework for studying the properties of samples, estimators, and inferences (the subjects of STAT 426 and STAT 427). This course will cover the axiomatic formulation of probability and formalize familiar concepts, including basic probability rules, random variables, distributions, and expectations. The course will also introduce concepts that play central roles in statistics, such as joint distributions, transformations, and conditional expectations.

Read more in the [course syllabus].

Announcements

Course-related:

• HW5 due 10/14/24 11:59pm PDT
• HW6 due 10/16/24 11:59pm PDT
• Quiz 1 corrections due 10/18/24 11:59pm PDT
• Late HW4 submissions due 10/14/24 11:59pm PDT
• Late HW5 submissions due 10/21/24 11:59pm PDT

Non-course-related:

• BCSM career fair Thursday 10/17 & Friday 10/18 [info]
• Statistics Frost SURP symposium Friday 10/18 11:10am – 12:00pm in 53-202

Instructor: Trevor Ruiz (he/him/his) [email] [website]

Learning assistant: Edy Reynolds [email]

Class meetings:

• [Section 01] 12:10pm – 2:00pm MW 186-C300
• [Section 02] 2:10pm – 4:00pm MW 186-C300

Office hours (OH) and learning assistant hours (LAH):

• [OH] M 4:00pm – 5:30pm [by appointment]

• [LAH] T 11:00am — 12:00pm 25-107G or [Zoom]

• [OH] W 9:30am – 11:00am [by appointment]

• [LAH] R 11:00am — 12:00pm 25-107G or [Zoom]

For office hours, drop ins are welcome (25-236) but availability is not guaranteed without an appointment. For LA hours, availability is drop-in only on a first come, first served basis.

Week 1 (9/23/24)

Set theory; probability axioms

Monday: sets and set operations [notes]

• [reading] sections 1.1, 1.2, 1.2.1
• [assignment] exercises 1.2.1, 1.2.2, 1.2.6 [solutions]
• [optional] exercises 1.2.5, 1.2.7

Wednesday: probability axioms and properties [notes]

• [reading] section 1.3
• [assignment] exercises 1.3.6, 1.3.7, 1.3.21, 1.3.22 [solutions]
• [optional] exercises 1.3.3, 1.3.5

Week 2 (9/30/24)

Discrete probability spaces; counting methods

Monday: counting rules [notes]

• [reading] section 1.3.1
• [assignment] exercises 1.3.10, 1.3.18, 1.3.19 [solutions]
• [optional] exercises 1.3.14, 1.3.15, 1.3.16

Wednesday: the matching problem [notes]

• [reading] section 1.3.2
• [assignment] exercises 1.3.11, 1.3.14 [solutions]

Week 3 (10/7/24)

Monday: Quiz 1 [clean copy]

• no reading or assignment

Wednesday: conditional probability; Bayes’ theorem [notes]

• [reading] section 1.4
• [assignment] exercises 1.4.2, 1.4.6, 1.4.8, 1.4.34
• [optional] exercises 1.4.4, 1.4.26

Week 4 (10/14/24)

Monday: the Monte Hall problem; independence [notes] [R script: sensitivity and specificity]

• [reading] section 1.4.1
• [assignment] exercises 1.4.11, 1.4.21, 1.4.31
• [optional] exercises 1.4.12, 1.4.19, 1.4.23, 1.4.26

Wednesday: the Gambler’s ruin problem; random variables

• [reading] section 1.5 through remark 1.5.1 (p. 39)
• [assignment] exercises 1.4.18, 1.4.13, 1.4.22, 1.4.32