MA 464 Theory of probability, Overview

 Thursday 1/16 Introduction: Bertrands paradox asks for a solid foundation of probability theory. The Petersburg Casino will be treated later. We will flip coins more often in this course. Tuesday 1/21 Kolmogorov's Axioms of Probability theory . Basic properties. Examples: 2 dices, Boys/Girl problem Bertrand solved. Homework 1 (PS file) Thursday 1/23 Examples: radioactive decay, Monty Hall problem Continuity property of P, Conditional probability: boy or girl problem revisited. An urn problem. Tuesday 1/28 The urn problem, Bayes rule, the drawer problem, independence of events. Pairwise independence is not enough. Homework 2 (PS file) Thursday 1/30 More examples on independence and conditional probability and Bayes rule. Theory: Bernoulli formula and Sylvester's "switch-on switch off" formula for the probability of a union of events. Reminder: even without probability some questions can be difficult: Life also can be complicated Tuesday 2/3 Review of combinatorics Permutations, Sampling and Combinations. Examples, the Bursday paradox. Distribution of Homework 3 (PS file) Thursday 2/5 More examples in combinatorics. Especially the odds in Arizona lotto and some combinatorial questions in music . Tuesday 2/11 Discrete random variables, densities, distributions, examples of distributions. How to get onto mars using probability. Homework 4 (PS file) Thursday 2/12 More examples of distributions : A sailor, Euler and a devilish random variable . Independent random variables. Tuesday 2/18 On distributions: what happens when adding independent random variables. From Bernoulli to Poisson. Applications. Checklist for Midterm. distributions II : Homework 5 (PS file) Thursday 2/20 Midterm topics: Material until Tuesday 2/18. Midterm 1 (PS file) Tuesday 2/25 Review over Midterm. Expectation of random discrete variables: Definition, computing examples. Expectation overview . Homework 6 (PS file) Thursday 2/27 Properties of expectation, more examples. Definition of variance. Probability generating functions and its use for computing expectation and variance. Tuesday 3/4 Covariance, correlation, correlation coefficient, regression line. Independent random variables are uncorrelated. Schwartz inequality. Homework 7 (PS file) Thursday 3/6 Excess and other higher moments, sum of independent random variables. Estimating probabilities with the Chebychev inequality. The weak law of large numbers for IID random variables. Tuesday 3/11 An application of the weak law of large numbers in analysis: the Weierstrass theorem We reviewed some material in form a Selftest (PS file) which is corrected by yourself and will not be part of the grade. The test should be useful for spotting eventual white spots in the first half of the course. Homework 8 (PS file) Thursday 3/13 Preparation for continuous random variables. Countable versus uncountable. Why did we introduce sigma algebras? The Banach-Tarski paradox. What is integration ? Hacking the code at the foundations of probability theory with the theorem of Caratheodory. The paradox of Schwartz on triangulations of surfaces. SPRING BREAK.
 Tuesday 3/25 Continuous random variables. Expectation for normal, uniform and Poisson distribution. Homework 9 Thursday 3/27 Second midterm. Midterm 2 Tuesday 4/1 The construction of random variables with a given distribution. How do densities transform? Some applications. Homework 10 Some continuous distributions. Thursday 4/3 Short repetition Chapter 5. Start with multidimensional random variables, joint and marginal distributions. The normal distribution in arbitrary dimensions. Random vectors (I) Homework 11 Random vectors (II) Tuesday 4/8 The distribution of a sum of independent random variables. Thursday 4/10 Change of variables for joint densities, Conditional densities. Tuesday 4/15 The central limit theorem. Lindeberg's central limit theorem with proof Homework 12 Thursday 4/17 Applications interpretation of the central limit theorem and the Poisson limit theorem. For example, a derivation of Stirlings formula or the interpretation of the central limit theorem as a fixed point theorem for a dynamical system. Some inequalities Tuesday 4/22 Characteristic functions. Recover of distribution from characteristic functions. CF for sum of independent random variables. Proof of the central limit theorem in the IID case. Homework 13 Thursday 4/24 A word on convergence of random variables. Moment generating functions and characteristic functions. Interpretation as results in Fourier theory. Calculation in some cases. Computing of moments. Characteristic functions for random vectors. Characteristic functions Convergence of random variables Thuesday 4/29 Random Walks: in which dimensions does a random walker return? The answer gives Polya's theorem . Gambling systems, can one beat the system? The Wald identities and an application for the Ruin probability of two gamblers The Borel-Cantelli lemma was used two times already in proofs. Homework 14 (Last one, really!) Thursday 5/1 Repetition random walk. More on the one dimensional random walk. Poisson Processes. Tuesday 5/3 Summary of the course and an outlook and remark on the final . Definition and existence of Brownian motion. The Black-Scholes formula in economics. The Feynman-Kac formula in quantum mechanics. The final

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