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Introduction to Probabilities

Introduction to Probabilities. Fall 2010 Dept. Electrical Engineering National Tsing Hua University 劉奕汶. What is probability ?. Literally, how probable an event is to occur. We live in a random world Relative-frequency interpretation 機 率 ／ 概 率 ／或然 率 This interpretation is problematic

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Introduction to Probabilities

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  1. Introduction to Probabilities Fall 2010 Dept. Electrical Engineering National Tsing Hua University 劉奕汶

  2. What is probability? • Literally, how probable an event is to occur. • We live in a random world • Relative-frequency interpretation • 機率／概率／或然率 • This interpretation is problematic • Involved law of large number • Not all experiments could be repeated • Not all repeating processes have convergent frequency • Axiomatic approach

  3. A bit of History • 3500 B.C., Egyptians used bones to gamble • Since then, dice, playing cards, mahjong, etc. • 15-16th centuries: Italy (Galilei et al.) • 17-18th centuries: Western-central Europe • Pascal, Fermat, Laplace, Poisson, Gauss • Huygens (1629-1695) On Calculations in Games of Chance • 19-20th : Russia • 1900: Hilbert’s 23 problems • 1933: Kolmogorov: probability theory axiomatized

  4. Probability in EE/CS • Signal processing • “Signal” = Random Process • Random because of noise and uncertainty • Machine learning • Natural language processing • Pattern recognition • Communication • Source coding • Channel coding • Modulation and estimation

  5. Probability in Finance/Economics • Investment / Gambling • Portfolio theory • Advertisement / Pricing

  6. Probability in Physics (i) • Statistical mechanics • Equilibrium • Entropy and 2nd law of thermodynamics • Definition of temperature

  7. Probability in Physics (ii) • Quantum mechanics • Schrödinger’s wave function • “Measurement makes reality” • The paradox of Schrödinger’s cat • Einstein’s famous comment

  8. Probability in Biomedicine • Genomics • Proteomics • Neuroscience • Ecology • Epidemiology

  9. Probability and Statistics • Law of Large Number • Central Limit Theorem • Why Gaussian distribution is “Normal” • Counter-example: stock market

  10. Syllabus • Textbook: S. Ghahramani, Fundamentals of Probability: with stochastic processes, 3rd Edition • Chapters 1-3: probability space • Chapters 4-5: discrete random variables • Chapter 6: Continuous random variables • Midterm exam (35%) • Chapters 7: continuous random variables II • Chapters 8: bivariate distributions • Chapter 10-11: advanced topics (Correlations, LLN, CLT, etc) • * Measure theory and axioms of probability • Final exam (35%) • A4 double-side cheat sheet permitted for both exams • 6 homework assignments (30%) • Office hours: Monday 5-6 pm, Rm 704B • Website: http://www.ee.nthu.edu.tw/ywliu/ee3060/

  11. Statistics of last semester’s grades (N = 37) • 期中考:M=51.4,SD=7.9 • 期末考:M=49.3,SD=10.8 • 總成績:M = 78,SD=11 • 36 passed, 1 failed. • 4 scored 90 or above (A+)

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