Boston University · Summer 2026

CAS MA 581 Probability

Instructor: Wancheng Lin

A six-week introduction to probability as a working language for counting, uncertainty, random variables, distributions, transformations, and limit theorems, with examples connecting classical problems to data science, economics, and finance.

Tentative Schedule

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.

Textbook and Recommended Resources

Sheldon Ross, A First Course in Probability, 10th edition, Pearson, 2019.

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.

Lecture Notes

Lecture notes will be posted here as the course begins. I will update this area in time.

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 HW problem sets.
Test 1 20% Wednesday, July 15. Closed book and closed note.
Test 2 20% Tuesday, July 28. Closed book and closed note.
Test 3 20% Thursday, August 6. Comprehensive.
Show-Up Bonus +10 max Added directly to your final calculated average.

Cheat sheet rule: you are permitted one double-sided A4 or letter-sized sheet of handwritten or typed notes/formulas for all exams.

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

Note: The schedule below is tentative and subject to minor adjustments as the summer term progresses. All readings correspond to Ross, 10th edition.

Week 1: Combinatorial Analysis, Axioms & Conditional Probability (June 29-July 2)

Homework: HW 1 released Thursday, July 2.

  • Lecture 1 (Mon, June 29): Combinatorial Foundations (Ch. 1.1-1.6)Course Introduction, Generalized Principle of Counting, Permutations, Combinations, Multinomial Coefficients, and the Number of Integer Solutions.Counting explains why rare outcomes can still be expected in huge spaces.
  • Lecture 2 (Tue, June 30): The Axioms of Probability & Equally Likely Outcomes (Ch. 2.1-2.5)Sample Spaces and Events, Axioms of Probability, Basic Propositions, and Sample Spaces Having Equally Likely Outcomes.A model is only as clear as its sample space.
  • Lecture 3 (Wed, July 1): Advanced Axioms & Intro to Conditional Probability (Ch. 2.6-2.7, 3.1-3.2)Probability as a Continuous Set Function, Probability as a Measure of Belief, and Introduction to Conditional Probabilities.Limits help turn many small events into stable probability rules.
  • Lecture 4 (Thu, July 2): Bayes's Formula & Conditioning Rules (Ch. 3.3, 3.5)The Multiplication Rule, Bayes's Formula, and conditional probability as a probability measure.Bayes tells us how evidence should update belief.

Week 2: Conditional Probability, Independence & Discrete RVs (July 6-July 9)

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

  • Lecture 5 (Mon, July 6): Independent Events and Conditional Independence (Ch. 3.4)Independence lets us simplify large systems into smaller parts.
  • Lecture 6 (Tue, July 7): Conditional Probability as a Measure; Introduction to Random Variables (Ch. 3.5, 4.1)Conditioning means recalculating probability with new information.
  • Lecture 7 (Wed, July 8): Discrete Random Variables, PMFs, and Expected Value (Ch. 4.2-4.3)Expected value is the long-run center of a random payoff.
  • Lecture 8 (Thu, July 9): Expectation of a Function of a Random Variable, Variance, and Identities (Ch. 4.4-4.5)Variance explains why equal averages can carry different risk.

Week 3: Discrete Distributions & Intro to Continuous RVs (July 13-July 16)

Homework: HW 2 due and HW 3 released Thursday, July 16.

  • Lecture 9 (Mon, July 13): Bernoulli, Binomial, and Poisson Random Variables (Ch. 4.6-4.7)Standard discrete laws model counts, arrivals, and rare events.
  • Lecture 10 (Tue, July 14): Geometric, Negative Binomial, Hypergeometric, Zeta Distributions, and CDF Properties (Ch. 4.8-4.10)Different sampling rules create different discrete distributions.
  • Lecture 11 (Wed, July 15): TEST 1 (In Class, 1 PM-2 PM) / Continuous Random Variables & Expectation (Ch. 5.1-5.2)Densities replace point probabilities for continuous quantities.
  • Lecture 12 (Thu, July 16): Uniform and Normal Random Variables, DeMoivre-Laplace Limit Theorem (Ch. 5.3-5.4)Many small independent effects often add up to normal-looking noise.

Week 4: Continuous Distributions & Jointly Distributed RVs (July 20-July 23)

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

  • Lecture 13 (Mon, July 20): Exponential, Gamma, Weibull, Cauchy, Beta, and Pareto Distributions; Hazard Rates (Ch. 5.5-5.6)Continuous distributions model waiting times, lifetimes, and heavy tails.
  • Lecture 14 (Tue, July 21): Distribution of a Function of a Random Variable (Ch. 5.7)Change of variables tracks how probability moves under transformations.
  • Lecture 15 (Wed, July 22): Joint Distribution Functions, Independent Random Variables (Ch. 6.1-6.2)Joint distributions describe how random quantities vary together.
  • Lecture 16 (Thu, July 23): Sums of Independent Random Variables and Order Statistics (Ch. 6.3, 6.6)Sums model totals; order statistics model best, worst, and extremes.

Week 5: Conditional Distributions & Properties of Expectation (July 27-July 30)

Homework: HW 4 due and HW 5 released Thursday, July 30.

  • Lecture 17 (Mon, July 27): Conditional Distributions (Discrete & Continuous), Joint Transformations, and Exchangeability (Ch. 6.4-6.5, 6.7-6.8)Conditional distributions update predictions after observing another variable.
  • Lecture 18 (Tue, July 28): TEST 2 (In Class, 1 PM-2 PM) / Expectation of Sums, Probabilistic Method, and Max-Min Identity (Ch. 7.1-7.2)Expectation can prove existence without constructing an example.
  • Lecture 19 (Wed, July 29): Moments of the Number of Events, Covariance, Variance of Sums, and Correlations (Ch. 7.3-7.4)Covariance explains when diversification helps.
  • Lecture 20 (Thu, July 30): Conditional Expectation, Prediction, and Computing by Conditioning (Ch. 7.5-7.6)Conditional expectation is the best average prediction after new information.

Week 6: Generating Functions & Limit Theorems (August 3-August 6)

Homework: HW 5 due Wednesday, August 5. No new homework assigned during final exam week.

  • Lecture 21 (Mon, Aug 3): Moment Generating Functions, Joint MGFs, Multivariate Normal, and Sample Mean/Variance (Ch. 7.7-7.8)MGFs package moments and identify many distributions.
  • Lecture 22 (Tue, Aug 4): Chebyshev's Inequality, Weak Law of Large Numbers, Poisson Approximations, and Chernoff Bounds (Ch. 8.1-8.5)Concentration bounds quantify uncertainty from limited data.
  • Lecture 23 (Wed, Aug 5): Central Limit Theorem, Strong Law of Large Numbers, Lorenz Curve, and Course Review (Ch. 8.3, 8.4, 8.7)Limit theorems explain stable averages; Lorenz curves show inequality.
  • Lecture 24 (Thu, Aug 6): TEST 3 (In Class, 1 PM-3 PM)