Probability Theory

Probability is the mathematics of random events. In statistics, probability theory provides a framework for studying the properties of samples, estimators, and inferences. This course will formally develop familiar probability rules from the Kolmogorov axioms and articulate core concepts related to random variables and their distributions, including distribution functions, expectations, and transformations.

Read more in the [course syllabus].

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

Class meetings:

• [Section 01] 2:10pm – 4:00pm TR 186-C200
• [Section 02] 4:10pm – 6:00pm TR 186-C200

Office hours:

• TR [by appointment] in 25-236 or via Zoom

Drop ins are welcome but availability is not guaranteed without an appointment.

Week 0 (9/18)

Thursday: probability axioms [notes]

• [reading] as needed for review: 1.2, 1.2.1, 1.2.2, 1.3.1
• [hw0] 1.2.1, 1.2.2, 1.3.1, 1.3.2, 1.3.4, 1.3.10, 1.3.18 [solutions]

This first assignment is intended to gauge the need for review of STAT305 material and will be “graded” based on attempted solutions only (not correctness).

Week 1 (9/22)

Tuesday: probability properties; a few counting problems [notes]

• [reading] 1.3, 1.3.2
• [hw1] 1.3.6, 1.3.7, 1.3.17, 1.3.22 [solutions]

Thursday: conditional probability [notes]

• [reading] 1.4, 1.4.1
• [hw2] 1.4.6, 1.4.8, 1.4.18, 1.4.21, 1.4.32, 1.4.34 [solutions]

Week 2 (9/29)

Tuesday: independence [applet] [notes]

• [reading] 1.4.1
• [hw3] 1.4.18, 1.4.21, 1.4.32 [solutions]

Thursday: random variables [notes]

• [reading] 1.5
• [hw4] 1.5.2, 1.5.4, 1.6.4, 1.6.7 [solutions]

Week 3 (10/6)

Tuesday: quiz 1 [scores]

• [optional practice] 1.3.13, 1.4.10, 1.4.15, 1.4.26, 1.5.7, 1.5.9

Thursday: cumulative distribution functions [applet] [notes]

• [reading] 1.6
• [hw5] 1.5.3, 1.7.15, sketch $Q(p)$ for each part in 1.5.4 [solutions]

Week 4 (10/13)

Tuesday: probability density functions [notes]

• [reading] 1.7, 1.7.2
• [hw6] 1.7.6, 1.7.12 (skip the median and 25th percentile), 1.7.14, 1.7.23 [solutions]

Thursday: expected value [notes]

• [reading] 1.8, 1.9
• [hw7] 1.7.26, 1.8.4, 1.8.7, 1.9.3, 1.9.27

Week 5 (10/20)

Tuesday: Gaussian distribution; probability inequalities

• [reading] 3.4, 1.10
• [hw8] 1.9.8, 3.4.1, 3.4.11, 1.10.1

Thursday: quiz 2

• [optional practice] 1.6.1, 1.6.9, 1.7.8, 1.7.17, 1.7.24, 1.9.20, 1.9.21, 1.9.22