slides1 - Lecture Stat 302 Introduction to Probability...

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Lecture Stat 302 Introduction to Probability - Slides 1 AD Jan. 2010 AD () Jan. 2010 1 / 18
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Administrative details Arnaud Doucet, O¢ ce LSK 308c, Department of Statistics & O¢ ce ICCS 189, Department of Computer Science. O¢ ce Hour, Department of Statistics: via appointment. Textbook: A °rst course in probability , 8th edition by Sheldon Ross. Website: http://people.cs.ubc.ca/~arnaud/stat302.html AD () Jan. 2010 2 / 18
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Notes, exercises and exams Notes posted on the web + Additional material. Exercises & solutions from textbook will be posted weekly. 3 assignements (20%). 1 mid-term (30%). 1 °nal exam (50%). AD () Jan. 2010 3 / 18
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Introduction to Probability Probability theory yields mathematical tools to deal with uncertain events. Used everywhere nowadays and its importance is growing. Applications include Population genetics: tree-valued stochastic processes, Web search engine: Markov chain theory, Data mining, Machine learning: Stochastic gradient, Markov chain Monte Carlo, Image processing: Markov random °elds, Design of wireless communication systems: random matrix theory, Optimization of engineering processes: simulated annealing, genetic algorithms, Computer-aided design of polymers: Markov chain Monte Carlo. Finance (option pricing, volatility models): Monte Carlo, dynamic models, Design of atomic bomb (Los Alamos): Markov chain Monte Carlo. AD () Jan. 2010 4 / 18
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Plan of the Course - Tentative Schedule 1 Combinatorial analysis; i.e counting (1 week) 2 Axioms of probability (1 week) 3 Conditional probability and inference (1 week) 4 Discrete & continuous random variables (2 weeks) 5 Multivariate random variables (1 week) 6 Properties of expectation, generating function (2 weeks) 7 Limit theorems: SLLN, CLT, inequalities (2 weeks) 8 Additional topics: Poisson and Markov processes (1 week) 9 Simulation and Monte Carlo methods (1 week) AD () Jan. 2010 5 / 18
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Combinatorial analyses aka Counting Many basic probability problems are counting problems. Example : Assume there are 1 man and 2 women in a room. You pick a person randomly. What is the probability P 1 that this is a man? If you pick two persons randomly, what is the probability P 2 that these are a man and woman Answer : You have the possible outcomes: (M), (W1), (W2) so P 1 = # ±successful² events # events = # boys # boys + # girls = 1 3 . To compute P 2 , you can think of all the possible events: (M,W1), (M,W2), (W1,W2) so P 2 = # ±successful² events # events = 2 3 .
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