It will be expected that the candidate knows material from a standard undergraduate, post-calculus level course in probability. Hence it is expected that the candidate knows the following:
Basic notions of probability and conditional probability including Bayes rule
Discrete Probability Densities (binomial, Poisson, geometric, hypergeometric)
Continuous Probability Densities (normal, exponential, uniform, joint normal)
Expectation, Variance, Standard Deviation, Covariance
Poisson approximation to binomial and normal approximations (central limit theorem)
Chebyshev's inequality and Weak Law of Large Numbers
For the exam, the student can choose one of two tracks. Track I consists primarily of measure theoretic probability and corresponds to Mathematics 287. Track II consists primarily of stochastic processes from a non-measure theoretic perspective and corresponds to Mathematics 216.
Measure theoretic foundations of probability theory :what is a probability space?; -algebras; random variables as measurable functions; notions of convergence (almost sure versus in probability)
Finite Markov chains in discrete time (recurrence classes, periodicity, convergence to invariant probability)
Borel-Cantelli Lemma; Zero-One Law
Expectation and other moments; convergence in ; weak law of large numbers ; strong law of large numbers (with idea of proof) ; law of iterated logarithm (without proof)
weak convergence of probability measures; characteristic functions of random variables and their relationship to weak convergence; central limit theorem with proof (in i.i.d. case)
conditional expectation; martingales (in discrete time); optional sampling theorem; martingale convergence theorem
Definition of Brownian motion; how is it constructed?; non-differentiability; strong Markov property
Markov chains with infinite state space: positive recurrence, null recurrence transience; simple random walk; reversible Markov chains and relationship between eigenvalues and convergence to equilibrium
Markov chains with continuous time: generator, relationship to discrete time models; examples: Poisson processes, Markovian queues
Branching processes
conditional expectation; martingales (in discrete time); optional sampling theorem; martingale convergence theorem
Brownian motion and stochastic integration from a non-measure theoretic perspective; Ito's formula
Tue Aug 26 17:27:18 EDT 1997