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

WEEK # | TOPICS |
---|---|

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) |

### Recitations

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) | (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) | (PDF) |

R22 | Central Limit Theorem (PDF) | (PDF) |

R23 | Last Recitation, Review Material Covered after Quiz 2 (Chapters 5-7) |

### Tutorials

SES # | Tutorials | SOLUTIONS |
---|---|---|

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 (PDF) | (PDF) |