Probability and Random Processes

Poisson Puffin

Spring 2023
Thomas Courtade

Lecture: Tue & Thu 12:30 pm - 2:00 pm, Lewis 100
Office Hour: Mon 1:00 pm - 2:00 pm, Cory 265


Lecture Schedule

Schedule is subject to some changes.

Date Topics Readings
1/17 Introduction/Logistics, Probability Basics B&T 1, W Appendix A
1/19 Bayes Rule, Independence, Discrete Random Variables B&T 1,2, W Appendix A
Random Variables
1/24 Expectation (Linearity, Tail Sum), Discrete Distributions B&T 2, W Appendix B
1/26 Sum of Independent Binomials, Variance, Covariance, Correlation Coefficient, Conditional Expectation and Iterated Expectation, Entropy B&T 2, W Appendix B
1/31 Entropy, Continuous Probability (Sample Space, Events, PDFs, CDFs), Continuous Distributions B&T 3, W Appendix B
2/2 Gaussian Distribution, Derived Distributions, Continuous Bayes B&T 3, 4.1-4.2, W Appendix B
2/7 Order Statistics, Convolution, Moment Generating Functions B&T 4.3-4.6, W Sections 3-4
2/9 MGFs, Bounds/Concentration Inequalities (Markov, Chebyshev, Chernoff) B&T 5.1, W Sections 3-4
Concentration Inequalities
2/14 Convergence, Weak and Strong Law of Large Numbers, Central Limit Theorem B&T 5.2-5.6, W Sections 3-4
2/16 No Lecture (Midterm 1)
2/21 Information Theory and Digital Communication, Capacity of the Binary Erasure Channel (BEC) W Section 7
Information Theory
2/23 Achievability of BEC Capacity, Markov Chains Introduction B&T 7.1-7.4l, W Section 7
Capacity of BEC
2/28 Discrete Time Markov Chains: Irreducibility, Aperiodicity, Invariant Distribution and Balance Equations B&T 7.1-7.4, W Sections 1-2
Discrete Time Markov Chains
3/2 DTMCs: Hitting Time and First Step Equations (FSEs), Infinite State Space, Classification of States, Big Theorem B&T 7.1-7.4, W Sections 1-2
3/7 DTMCs: Classification, Reversibility, Poisson Processes: Construction B&T 6.1-6.3, W Sections 1-2
3/9 Poisson Processes: Counting Process, Memorylessness, Merging, Splitting B&T 7.5, W Section 5
Poisson Processes
3/14 Poisson Processes: Erlang Distribution, Random Incidence, Continuous Time Markov Chains Introduction, Rate Matrix B&T 7.5, W Section 5
3/16 CTMCs: Balance Equations, Big Theorem, FSEs B&T 7.5, W Section 6
Continuous Time Markov Chains
3/21 CTMCs: Simulated DTMC, Erdos-Renyi Random Graphs W Section 6
Random Graphs
3/23 No Lecture (Midterm 2)  
3/28 No Lecture (Spring Break)  
3/30 No Lecture (Spring Break)  
4/4 Maximum Likelihood Estimation, Maximum A Posteriori Estimation B&T 8.1-8.2, 9.1
4/6 MLE/MAP, Neyman Pearson Hypothesis Testing B&T 8.1-8.2, 9.3-9.4, W Sections 7-8
Hypothesis Testing
4/11 Neyman Pearson Hypothesis Testing, Vector Space of Random Variables and Least Squares Estimation B&T 9.3-9.4, 8.3-8.5, W Sections 8-9
Hilbert Space of RVs
4/13 Linear Least Squares Estimation, Minimum Mean Square Error (MMSE) Estimation B&T 8.3-8.5, W Section 9
4/18 MMSE, Gram Schmidt Process W Sections 9-10
4/20 Jointly Gaussian Random Variables, Kalman Filter W Sections 8, 10
Jointly Gaussian RVs
4/25 Kalman Filter W Section 10
Kalman Filter (1)
Kalman Filter (2)
4/27 Hidden Markov Models W Section 11
Hidden Markov Models