Random Processes in Systems

Week 1

Jan 16

Probability Spaces; Kolmogorov Axioms; Consequences

1.1, App. A

Jan 17

Section Discussion 1

Jan 18

Borel-Cantelli; Existence/Uniqueness of Probability; Random Variables

1.2.1-1.2.4; review 1.2.7 on your own

Jan 21

HW 1 due

Week 2

Jan 23

Random Variables; Joint Distributions; Independence

1.2.3-1.2.4

Jan 24

Section Discussion 2

Jan 25

Expectation; Fatou’s Lemma; Monotone and Dominated Convergence Theorems

1.2.5-1.2.6

Week 3

Jan 29

HW 2 due

Jan 30

No Lecture

Jan 31

Section Discussion 3

Feb 1

Conditional Probability

1.3

Week 4

Feb 5

HW 3 due

Feb 6

Limit Theorems; Stochastic Processes

1.5, 2.1-2.2

Feb 7

Section Discussion 4

Feb 8

Strong Law of Large Numbers

2.1-2.2

Week 5

Feb 12

HW 4 due

Feb 13

Modes of Convergence; Central Limit Theorem

2.3.1-2.3.2

Feb 14

Section Discussion 5

Feb 15

Gaussian Vectors; Central Limit Theorem

2.3.1-2.3.2, Ch. 3

Week 6

Feb 19

HW 5 due

Feb 20

Gaussian Processes; Central Limit Theorem

2.3.1-2.3.2, Ch. 3

Feb 21

Section Discussion 6

Feb 22

Linear Estimation; Hilbert Space; Best Linear Estimator

App. B; 4.1-4.2

Week 7

Feb 26

HW 6 due

Feb 27

More on Linear Estimation; Linear Innovations; Kalman Filter

4.3-4.5

Feb 28

Section Discussion 7

Feb 29

No Lecture

Week 8

Mar 5

Midterm

Second-order Processes; Spectral Theory; Wide Sense Stationarity

5.1-5.3; App. C

Mar 6

Section Discussion 8

Mar 7

Optimal Filtering of WSS Processes

5.4-5.5

Week 9

Mar 11

HW 8 due

Mar 12

No Lecture

Mar 13

Section Discussion 9

Mar 14

WSS Signals and LTI Systems

5.5

Week 10

Mar 18

HW 9 due

Mar 19

Discrete-Time Markov Chains; Chapman-Kolmogorov Equations; Classification of States

6.1

Mar 20

Section Discussion 10

Mar 21

Discrete-Time Markov Limit Theorems

6.2

Week 11

Spring Break!

Week 12

Apr 1

HW 10 due

Apr 2

Reversibility; Spectral Gap

6.3

Apr 3

Section Discussion 11

Apr 4

Martingales - Definitions and Examples

7.1

Week 13

Apr 8

HW 11 due

Apr 9

Stopping Times; Optional Stopping Theorem

7.2

Apr 10

Section Discussion 12

Apr 11

Concentration Inequalities; Martingale Convergence Theorem

7.3-7.4

Week 14

Apr 15

HW 12 due

Apr 16

Poisson Processes - Definitions and Properties

Ch. 8

Apr 17

Section Discussion 13

Apr 18

Continuous-Time Markov Chains; Kolmogorov Differential Equations; Classifcation of States

9.1-9.2

Week 15

Apr 22

HW 13 due

Apr 23

Continuous-Time Markov Limit Theorems

9.3

Apr 24

Section Discussion 14

Apr 25

Reversibility and Spectral Gap

9.4