This class gives an introduction to theory and application of stochastic processes, including martingales, Brownian motion, Poisson processes, and point processes.
Review of discrete probability, see notes
Introduction to proability theory: axioms of discrete probability, conditional probability (HJE 5.1, 5.2.1, 5.2.2)
Law of total probability, Bayes’ rule, independence, discrete random variables, expectation (HJE 5.2.3, 5.2.4, 5.3, 5.4 in parts)
In class discussion, Exercise 1
Expectation, variance, Bernoulli trials, binomial distribution (HJ 5.4, 5.5.1, 5.5.2)
Continuous probability distributions (HJ 5.6.1), uniform and normal distributions (parts of HJ 5.6.2)
The Markov inequality, the Chebyshev inequality, and the weak law of large numbers (HJ 6.2); moment generating functions; the Poisson distribution as a limit of the Binomial distribution (see HJ 5.5.3 for the Poisson distribution, Section 4 of this set of lecture notes for the limit via moment generating functions)
Cumulants and cumulant generating functions, the central limit theorem (see HJ 6.3.1 for the central limit theorem, this set of lecture notes for a sketch of a proof via cumulants)
Discussion of Exercise 2
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