# Probability and Random Processes

Spring 2021
TuTh 2-3:30 PM, Internet/Online

Office Hours: Wednesday 12-1 PM

## Announcements

• Welcome to EECS 126! Please read the course info, join Piazza, and join Gradescope (code RWKVJ7).

## Lecture Schedule

Readings refer to Walrand’s “Probability in Electrical Engineering and Computer Science”. Online notes only serve as optional supplemental readings, and will not directly correspond to the lectures or textbook (see content). The B&T textbook may also be useful but is not the primary textbook.

Schedule is subject to some changes.

1/19 Introduction/Logistics, Probability Basics Appendix A, B&T 1
1/21 Bayes Rule, Independence, Discrete Random Variables Appendix A, B&T 1,2
1/26 Expectation (Linearity, Tail Sum), Discrete Distributions Appendix B, B&T 2
1/28 Sum of Independent Binomials, Variance, Covariance, Correlation Coefficient, Conditional Expectation and Iterated Expectation, Entropy Appendix B, B&T 2
2/2 Entropy, Continuous Probability (Sample Space, Events, PDFs, CDFs), Continuous Distributions Appendix B, B&T 3
2/4 Gaussian Distribution, Derived Distributions, Continuous Bayes Appendix B, B&T 3, 4.1-4.2
2/9 Order Statistics, Convolution, Moment Generating Functions Sections 3-4, B&T 4.3-4.6
2/11 MGFs, Bounds/Concentration Inequalities (Markov, Chebyshev, Chernoff) Sections 3-4, B&T 5.1
2/16 Convergence, Weak and Strong Law of Large Numbers, Central Limit Theorem Sections 3-4, B&T 5.2-5.6
Convergence
2/18 No Lecture (Midterm 1)
2/23 Information Theory and Digital Communication, Capacity of the Binary Erasure Channel (BEC) Section 7
Capacity of a BEC
2/25 Achievability of BEC Capacity, Markov Chains Introduction Section 7, B&T 7.1-7.4
Information Theory
3/2 Discrete Time Markov Chains: Irreducibility, Aperiodicity, Invariant Distribution and Balance Equations Sections 1-2, B&T 7.1-7.4
Markov Chains
3/4 DTMCs: Hitting Time and First Step Equations (FSEs), Infinite State Space, Classification of States, Big Theorem Sections 1-2, B&T 7.1-7.4
3/9 DTMCs: Classification, Reversibility, Poisson Processes: Construction Sections 1-2, B&T 6.1-6.3
Reversibility
3/11 Poisson Processes: Counting Process, Memorylessness, Merging, Splitting Section 5, B&T 7.5
3/16 Poisson Processes: Erlang Distribution, Random Incidence, Continuous Time Markov Chains Introduction, Rate Matrix Section 5, B&T 7.5
3/18 CTMCs: Balance Equations, Big Theorem, FSEs Section 6, B&T 7.5
CTMCS
3/23 No Lecture (Spring Break)
3/25 No Lecture (Spring Break)
3/30 CTMCs: Simulated DTMC, Erdos-Renyi Random Graphs Section 6
Random Graphs
4/1 Maximum Likelihood Estimation, Maximum A Posteriori Estimation B&T 8.1-8.2, 9.1
4/6 No Lecture (Midterm 2)
4/8 MLE/MAP, Neyman Pearson Hypothesis Testing Sections 7-8, B&T 8.1-8.2, 9.3-9.4
Hypothesis Testing
4/13 Neyman Pearson Hypothesis Testing, Vector Space of Random Variables and Least Squares Estimation Sections 8-9, B&T 9.3-9.4, 8.3-8.5
Hilbert Space of RVs
4/15 Linear Least Squares Estimation, Minimum Mean Square Error (MMSE) Estimation Section 9, B&T 8.3-8.5
4/20 MMSE, Gram Schmidt Process Sections 9-10
4/22 Jointly Gaussian Random Variables, Kalman Filter Sections 8, 10
Kalman Filter
4/27 Kalman Filter Section 10
4/29 Hidden Markov Models Section 11
Hidden Markov Models