Applied Probability Qualifying Examination Syllabus

  1. Probability spaces, rules of probability, conditional probability, independent events,  distributions of random variables,  expectation, variance,  joint distributions,   covariance, and correlation,  multivariate normal distribution, transformations of random variables and random vectors, probability generating functions, moment generating functions, characteristic functions,  Markov inequality, Chebyshev inequality, Cauchy-Schwarz inequality, Jensen’s inequality,  laws of large numbers, central limit theorem.
  2. Conditional probability distributions, conditional expectation, conditional variance, use of conditioning to solve problems,   basic  definitions and properties of  Markov chains,  random walks,  Poisson processes,   renewal processes,  Brownian motion,  Gaussian processes,  martingales, and stationary processes.
  3. Basic concepts of measure theoretic probability: countable and uncountable sets,  measure spaces,   integrals with respect to a measure or a probability distribution,  convergence of random variables in distribution, in probability, almost surely, in mean square, relations between the modes of convergence of random variables.


Textbooks and references.

Probability, An Introduction by Geoffrey Grimett and Dominic Welsh
Introduction to Probability Models by Sheldon Ross
An Introduction to Stochastic Modeling by Mark A. Pinsky and Samuel Karlin
A First Look at Rigorous Probability Theory by Jeffrey S. Rosenthal, World Scientific Press.
Convergence of Random Variables,  Wikipedia.