# Probability and Random Processes

## Lecture Schedule

(*Lectures are not recorded)
01/22 Introduction/Logistics, Probability Spaces/Axioms, Conditional Probability, Multiplication Rule, Law of Total Probability, Bayes Rule B-T 1 Discussion 1 (Solutions)
1/24 Bayes Rule, Independence, Discrete Random Variables B-T 1,2 Homework 1 (Solutions)
Lab 1 (Solutions)
1/29 Expectation (Linearity, Tail Sum), Uniform, Geometric, Binomial and Poisson Distributions B-T 2 Discussion 2 (Solutions)
1/31 Sum of Independent Binomials, Variance, Covariance, Correlation Coefficient, Conditional Expectation and Iterated Expectation, Entropy B-T 2 Homework 2 (Solutions)
Lab 2 (Solutions)
2/5 Entropy, Continuous Probability (Sample Space, Events, PDFs, CDFs), Uniform, Exponential Distributions B-T 3 Discussion 3 (Solutions)
2/7 Gaussian Distribution, Derived Distributions, Continuous Bayes B-T 3, B-T 4.1-4.2 Homework 3 (Solutions)
Lab 3 (Solutions)
2/12 Order Statistics, Convolution, Moment Generating Functions B-T 4.3-4.6 Discussion 4 (Solutions)
2/14 MGFs, Bounds/Concentration Inequalities (Markov, Chebyshev, Chernoff) B-T 5.1, W 13.7 Homework 4 (Solutions)
2/19 No Lecture (Midterm) Discussion 5 (Solutions)
2/21 Convergence, Weak and Strong Law of Large Numbers, Central Limit Theorem B-T 5.2-5.6, W 2.1-2.3 Homework 5 (Solutions)
2/26 Central Limit Theorem Proof Sketch, Information Theory and Digital Communication, Capacity of the Binary Erasure Channel (BEC) Capacity of BEC Discussion 6 (Solutions)
2/28 Achievability of BEC Capacity, Markov Chains Introduction W 1, 13.3, B-T 7.1-7.4 Homework 6 (Solutions)
Lab 4 (Solutions)
3/5 Discrete Time Markov Chains: Irreducibility, Aperiodicity, Invariant Distribution and Balance Equations W 1, 2.4, 2.6, 13.3, B-T 7.1-7.4 Discussion 7 (Solutions)
3/7 DTMCs: Hitting Time and First Step Equations (FSEs), Infinite State Space, Classification of States, Big Theorem W 1, 2.4, 2.6, 13.3, B-T 7.1-7.4, Markov Chains Note Homework 7 (Solutions)
Lab 5 (Solutions)
3/12 DTMCs: Classification, Reversibility, Poisson Processes: Construction B-T 6.1-6.3, W 13.4, Reversibility Note Discussion 8 (Solutions)
3/14 Poisson Processes: Counting Process, Memorylessness, Merging, Splitting B-T 6.1-6.3, W 13.4 Homework 8 (Solutions)
3/19 Poisson Processes: Erlang Distribution, Random Incidence, Continuous Time Markov Chains Introduction, Rate Matrix B-T 7.5, W 13.5 Discussion 9 (Solutions)
3/21 CTMCs: Balance Equations, Big Theorem, FSEs B-T 7.5, W 13.5 Homework 9 (Solutions)
3/26 Spring Break
3/28 Spring Break
4/2 CTMCs: Simulated DTMC, Erdos-Renyi Random Graphs Random Graphs Discussion 10 (Solutions)
4/4 Maximum Likelihood Estimation, Maximum A Posteriori Estimation W 5.1, B-T 8.1-8.2, 9.1 Lab 6 (Solutions)
4/9 No Lecture (Midterm) Discussion 11 (Solutions)
4/11 MLE/MAP, Neyman Pearson Hypothesis Testing W 5.1, B-T 8.1-8.2, 9.1/ W 5.5-5.6, 6.5, B-T 9.3-9.4, Hypothesis Testing Homework 10 (Solutions)
4/16 Neyman Pearson Hypothesis Testing, Vector Space of Random Variables and Least Squares Estimation W 5.5-5.6, 6.5, B-T 9.3-9.4/ W 7.1-7.5, B-T 8.3-8.5 Discussion 12 (Solutions)
4/18 Linear Least Squares Estimation, Minimum Mean Square Error (MMSE) Estimation W 7.1-7.5, B-T 8.3-8.5, Hilbert Space of Random Variables Homework 11 (Solutions)
4/23 MMSE, Gram Schmidt Process W 7.1-7.5, W 8.1 Discussion 13(Solutions)
4/25 Jointly Gaussian Random Variables, Kalman Filter W 6.3-6.4, 7.6, 8.1-8.3 Geometric Derivation of Scalar Kalman Filter Homework 12(Solutions)
Lab 7(Solutions)
4/30 Kalman Filter W 7.6, 8.1-8.3 Discussion 14 (Solutions)
5/2 Hidden Markov Models W 9.2,9.4, HMMs and the Viterbi Algorithm Homework 13(Solutions )

## Discussions

Discussion worksheets will be posted here.

## Homework

Homework will be posted here.

## Labs

Labs will be posted here.