Boston University · Summer 2026

CAS MA 581 Probability

Instructor: Wancheng Lin

A six-week introduction to probability as a working language for set models, conditional reasoning, random variables, distribution families, expectation, transformations, simulation, inequalities, and limit theorems.

Tentative Schedule Lecture Packet

Course Information

Session
Summer 2, June 29-August 7, 2026
Section
B1 (IND)
Meeting Time
Monday, Tuesday, Wednesday, Thursday · 1-3 pm
Location
COM 215
Prerequisites
CAS MA 225 or CAS MA 230; or consent of instructor

Prerequisites

Undergraduate Prerequisites: CAS MA 225 or CAS MA 230; or consent of instructor.

Graduate Prerequisites: CAS MA 225 or CAS MA 230; or consent of instructor.

Course Description

Basic probability, conditional probability, independence. Discrete and continuous random variables, mean and variance, functions of random variables, moment generating function. Jointly distributed random variables, conditional distributions, independent random variables. Methods of transformations, law of large numbers, central limit theorem. Cannot be taken for credit in addition to CAS MA 381.

The course is intended for students who want a solid working foundation in probability for further study in statistics, stochastic processes, data science, economics, finance, and related quantitative fields. Exercises and practice problems may draw from classical combinatorial examples as well as applications in modern data science, economics, and finance.

Key Components From the Lecture Packet

1

Probability Models

Experiments, sample spaces, events, set operations, partitions, Kolmogorov axioms, and probability rules.

2

Conditioning

Conditional probability, total probability, Bayes' rule, independence, and conditional independence.

3

Discrete Random Variables

PMFs, CDFs, Bernoulli, binomial, hypergeometric, geometric, negative binomial, and Poisson models.

4

Joint Discrete Models

Joint, marginal, and conditional distributions; independence; expectation, variance, covariance, and conditioning identities.

5

Continuous Models

CDFs, PDFs, uniform, exponential, gamma, beta, normal distributions, tail probabilities, and memorylessness.

6

Transformations and Limits

Functions of random variables, inverse-CDF simulation, joint continuous variables, transformations, convolution, MGFs, inequalities, LLN, and CLT.

Textbook and Recommended Resources

Main text: Neil A. Weiss, A Course in Probability, Pearson, 2006.

The Random Services probability project is a useful supplementary resource for probability definitions, examples, simulations, and distribution reference material.

Additional references or short notes may be posted as the course develops.

Attendance & Eagle-Eye Bonus Policy

This summer session moves fast—consistency and sharp attention are rewarded. Earn up to +10 bonus points applied directly to your final grade:

Earn +2 points per week by meeting these expectations:

  • Actively attend all four lectures (Mon–Thu) each week. Any absence drops that week's bonus to 0.
  • Help develop and refine weekly lecture LaTeX code. Submit by Friday following that week. Quality and completeness required.
  • Point out typos, errors, or mistakes in lecture slides or materials. Include lecture date and description. Much appreciated!

Maximum: +10 points total (5 weeks × +2 points per week). You must meet all three expectations in a given week to earn that week's +2 points.

Final Grade & Exam Summary

Component Weight Exam Rules & Details
Homework 40% 5 problem sets tied to the weekly lecture packet sections. Lowest homework score dropped.
Test 1 20% Wednesday, July 15. Covers probability models, conditioning, and discrete random variables.
Test 2 20% Tuesday, July 28. Covers joint distributions, expectation, variance, and continuous random variables.
Test 3 20% Thursday, August 6. Comprehensive, with emphasis on transformations, MGFs, inequalities, LLN, and CLT.
Show-Up Bonus +10 max Added directly to your final calculated average.

Exam rule: tests are in class. You may use one double-sided A4 or letter-sized formula sheet for each exam unless announced otherwise.

Dropped grades: The lowest homework assignment will be dropped, and the lowest test score (from Test 1, Test 2, or Test 3) will be dropped from your final grade calculation.

Grading weights and schedule benchmarks are tentative and may be adjusted slightly depending on the collective progress of the class.

Generative AI Policy

In this course, we are committed to fostering a learning environment that embraces emerging technologies while upholding the core principles of academic integrity: honesty, trust, fairness, respect, and responsibility.

Generative Artificial Intelligence (AI) tools are permitted and encouraged for specific purposes, primarily as a personalized learning mentor and tutor, to enhance the learning process, not to replace it.

AI should never be used to simply generate submitted content such as assignment answers or code.

Tentative Course Schedule

The schedule is organized around the current lecture packet and the main text, Neil A. Weiss, A Course in Probability. It may be adjusted as the summer term progresses.

24 class meetings
5 homework sets
3 in-class tests
6 compressed weeks
Week 1

Probability Models, Set Laws & Counting

June 29-July 2

Milestone: HW 1 released Thursday, July 2.

L1. Experiments, sample spaces, and events Random experiments, outcomes, events as sets, complements, unions, intersections, and disjointness.
L2. Set algebra and probability axioms De Morgan's laws, partitions, Kolmogorov axioms, additivity, and probability rules.
L3. Counting and equally likely outcomes Product rules, permutations, combinations, multinomial coefficients, and finite sample spaces.
L4. Conditional probability and Bayes' rule Conditioning, total probability, Bayes' rule, generalized Bayes' rule, and evidence updates.
Week 2

Independence & Discrete Random Variables

July 6-July 9

Milestone: HW 1 due and HW 2 released Thursday, July 9.

L5. Independence and repeated trials Independent events, conditional independence, product rules, and Bernoulli trial language.
L6. Discrete random variables Ranges, PMFs, CDFs, transformed discrete variables, and distribution notation.
L7. Core discrete models I Bernoulli, binomial, hypergeometric, geometric, and negative binomial distributions.
L8. Core discrete models II Poisson random variables, rare-event modeling, and choosing the right discrete model.
Week 3

Joint Models, Expectation & Variance

July 13-July 16

Milestone: HW 2 due and HW 3 released Thursday, July 16. Test 1 on Wednesday.

L9. Joint discrete distributions Joint PMFs, marginals, conditional PMFs, and independence of random variables.
L10. Expectation, variance, covariance Expectation identities, variance formulas, covariance, correlation, and sums.
L11. Test 1 + conditional expectation Test 1, 1-2 pm. Then total expectation, prediction, conditional variance, and conditioning computations.
L12. From discrete to continuous CDF properties, probabilities as areas, PDFs, normalization, and PDF/CDF relationships.
Week 4

Continuous Distributions & Transformations

July 20-July 23

Milestone: HW 3 due and HW 4 released Thursday, July 23.

L13. Core continuous models Uniform, exponential, gamma, beta, normal, tail probabilities, and memorylessness.
L14. Functions and simulation CDF method, PDF change of variables, monotone transformations, and inverse-CDF simulation.
L15. Joint continuous distributions Joint PDFs/CDFs, marginal densities, conditional densities, and independence tests.
L16. Continuous expectation identities Expectation integrals, variance, covariance, correlation, conditional expectation, and total variance.
Week 5

Convolution, MGFs & Inequalities

July 27-July 30

Milestone: HW 4 due and HW 5 released Thursday, July 30. Test 2 on Tuesday.

L17. Bivariate transformations Jacobian method, convolution, sums of independent variables, ratios, and product distributions.
L18. Test 2 + moment generating functions Test 2, 1-2 pm. Then raw moments, MGFs, derivatives, and distribution identification.
L19. MGF properties and sums Uniqueness, affine transformations, independent sums, normal stability, and MGF convergence.
L20. Markov and Chebyshev inequalities Tail bounds from expectation and variance, useful versus loose bounds, and sample-average examples.
Week 6

Limit Theorems & Final Review

August 3-August 6

Milestone: HW 5 due Wednesday, August 5. Test 3 on Thursday.

L21. Law of large numbers Sample averages, convergence in probability, Chebyshev proof, and repeated-sampling interpretation.
L22. Central limit theorem Standardized sums, normal approximation, and applied probability calculations for totals.
L23. Comprehensive review Conditioning, distribution recognition, expectation identities, transformations, MGFs, inequalities, LLN, and CLT.
L24. Test 3 Comprehensive in-class test, 1-3 pm.