This section contains problems that are solved during recitation and tutorial sessions in addition to weekly notes that give an overview of topics to be covered. During recitations, the instructor elaborates on theories, solves new examples, and answers students' questions. Recitations are held separately for undergraduates and graduates. During tutorials, students discuss and solve new examples with a little help from the instructor. Tutorials are active sessions to help students develop confidence in thinking about probabilistic situations in real time. Tutorials are not mandatory but highly recommended for students enrolled in the course.

Weekly Notes

1 Probability Models and Axioms (PDF)
2 Conditional Probability and Baye's Rule (PDF)
3 Discrete Random Variables, Probability Mass Functions, and Expectations (PDF)
4 Conditional Expectation and Multiple Discrete RVs (PDF)
5 Continuous RVs (CDF, Normal RV, Conditioning, Multiple RV) (PDF)
6 Continuous RVs (Conditioning, Multiple RVs, Derived Distributions) (PDF)
7 Derived Distributions, Convolution, and Transforms (PDF)
8 Iterated Expectations, Sum of a Random Number of RVs (PDF)
9 Prediction, Covariance and Correlation, Weak Law of Large Numbers (PDF)
10 Weekly Notes
11 Bernoulli Process, Poisson Process (PDF)
12 Weekly Notes
13 Markov Chains (Steady State Behavior and Absorption Probabilities) (PDF)
14 Central Limit Theorem (PDF)



SES # Recitations SOLUTIONS
R1 Set Notation, Terms and Operators (include De Morgan's), Sample Spaces, Events, Probability Axioms and Probability Laws (PDF) (PDF)
R2 Conditional Probability, Multiplication Rule, Total Probability Theorem, Baye's Rule (PDF) (PDF)
R3 Introduction to Independence, Conditional Independence (PDF) (PDF)
R4 Counting; Discrete Random Variables, PMFs, Expectations (PDF) (PDF)
R5 Conditional Expectation, Examples (PDF) (PDF)
R6 Multiple Discrete Random Variables, PMF (PDF) (PDF)
R7 Continuous Random Variables, PMF, CDF (PDF) (PDF)
R8 Marginal, Conditional Densities/Expected Values/Variances (PDF) (This resource may not render correctly in a screen reader.PDF)
R9 Derivation of the PMF/CDF from CDF, Derivation of Distributions from Convolutions (Discrete and Continuous) (PDF) (PDF)
R10 Transforms, Properties and Uses (PDF) (PDF)
R11 Iterated Expectations, Random Sum of Random Variables (PDF) (PDF)
R12 Expected Value and Variance (PDF) (PDF)
R13 Recitation 13 (PDF)
R14 Prediction; Covariance and Correlation (PDF) (PDF)
R15 Weak Law of Large Numbers (PDF) (PDF)
R16 Bernoulli Process, Split Bernoulli Process (PDF) (PDF)
R17 Poisson Process, Concatenation of Disconnected Intervals (PDF) (PDF)
R18 Competing Exponentials, Poisson Arrivals (PDF) (PDF)
R19 Markov Chain, Recurrent State (PDF) (PDF)
R20 Steady State Probabilities, Formulating a Markov Chain Model (PDF) (PDF)
R21 Conditional Probabilities for a Birth-death Process (PDF) (This resource may not render correctly in a screen reader.PDF)
R22 Central Limit Theorem (PDF) (PDF)
R23 Last Recitation, Review Material Covered after Quiz 2 (Chapters 5-7)  



T1 Baye's Theorem, Independence and Pairwise Independence (PDF) (PDF)
T2 Probability, PMF, Means, Variances, and Independence (PDF) (PDF)
T3 PMF, Conditioning and Independence (PDF) (PDF)
T4 Expectation and Variance, CDF Function, Expectation Theorem, Baye's Theorem (PDF) (PDF)
T5 Random Variables, Density Functions (PDF) (PDF)
T6 Transforms, Simple Continuous Convolution Problem (PDF) (PDF)
T7 Iterated Expectation, Covariance/Independence with Gaussians, Random Sum of Random Variables (PDF) (PDF)
T8 Correlation, Estimation, Convergence in Probability (PDF) (PDF)
T9 Signal-to-Noise Ratio, Chebyshev Inequality (PDF) (PDF)
T10 Two Instructive Drill Problems (One Bernoulli, One Poisson) (PDF) (PDF)
T11 Poisson Process, Conditional Expectation, Markov Chain (PDF) (PDF)
T12 Markov Chains: Steady State Behavior and Absorption Probabilities (This resource may not render correctly in a screen reader.PDF) (PDF)