IERG 6300 Theory of Probability Lecture 1 Sept

IERG 6300 Theory of Probability Lecture 1 Sept 2018 kshum

Staff • Lecturer – Kenneth Shum – http: //home. ie. cuhk. edu. hk/~wkshum/ – Office: SHB 736 (inside Institute of Network Coding) – Office hour: Wednesday afternoon 3 pm~4 pm. • Teaching Assistant – Huang Huaiyi – Office: SHB 702 Sept 2018 kshum 2

A rigorous treatment of probability • Kolmogorov’s 1933 monograph on the foundation of probability theory – http: //www. mathematik. com/Kolmogorov/0000. html • This course covers the materials in a typical graduate math course on probability theory – Based on measure theory – This course significantly overlaps with MATH 5011 Real Analysis I • Roughly half of the course is about mathematical analysis. Sept 2018 kshum 3

What this course is NOT about • This is not a course on Bayesian reasoning or statistical learning, decision making, etc. – Please refer to IERG 5130 • We will not talk about statistical modeling – This requires estimation of system parameters, goodness-of-fit tests, etc. – IERG 3050 • In this course we seldom compute a probability value numerically. Sept 2018 kshum 4

Applications • Mathematical statistics – Large-sample theory • Random graphs and complex networks • Analysis of low-density parity-check codes – Martingale theory • Financial mathematics – Stochastic calculus, Ito integral, Brownian motion • Machine Learning Sept 2018 kshum 5

Textbook • “Probability Theory” by A. Klenke, 2014 • Freely downloadable form University Library • https: //www. springer. c om/gp/book/97814471 53603 Sept 2018 kshum 6

Grading • Biweekly Homework, 30% • Midterm exam, 20% • Final exam, 50% Sept 2018 kshum 7

Course webpage • piazza. com • Homework, solutions, and additional materials will be posted in piazza. com • You are welcome to ask questions in piazza – Anonymous mode is available. • I will put some questions related to the lecture from time to time. Sept 2018 kshum 8

Banach Taski paradox • https: //en. wikipedia. org/wiki/Banach%E 2%80%9 3 Tarski_paradox • https: //www. youtube. com/watch? v=Tk 4 ubu 7 Bl. Sk • https: //www. youtube. com/watch? v=s 86 -Z-Cba. HA Sept 2018 kshum 9

Jointly Gaussian random variables • Suppose X and Y are jointly Gaussian random variables – E[X] = 1 – E[Y] = 1 – E[XY] = , -1 < < 1 • The joint pdf is • What happen when approaches 1? Sept 2018 kshum 10