Suggested Texts

  • Ross, A First Course in Probability, 9th edition
  • Hogg, McKean and Craig, Introduction to Mathematical Statistics, 6th edition
  • Casella and Berger, Statistical Inference, 2nd edition
  • Durrett, Essentials of Stochastic Processes, 2nd edition

Syllabus

Probability Theory core material:

  • Basic probability:

    • Probability axioms, independence
    • Random variables, cumulative distribution functions, probability mass functions, probability density functions, joint distributions, expectation, variance
    • Binomial, geometric, Poisson, uniform, normal, and exponential distributions
    • Conditional probability, conditional distributions, conditional expectation
  • Limit theorems:

    • Modes of convergence (distribution, probability, almost sure, pth mean)
    • Weak and strong law of large numbers
    • Central limit theorem
    • Slutsky’s theorem

Mathematical Statistics core material:

  • Basics
    • Transformations of random variables
    • Multivariate transformations
    • Order statistics, minima and maxima
    • Moment generating functions, characteristic functions
    • Exponential families
  • Estimation
    • Bias, mean squared error
    • Method of moments
    • M-estimators
    • Maximum likelihood, asymptotic properties, invariance
    • Cramer-Rao lower bound
    • Statistical efficiency
    • EM algorithm
    • Uniformly minimum variance unbiased estimators
    • Sufficiency, completeness, Basu’s theorem
    • Rao-Blackwell theorem
    • Lehmann-Scheffe theorem
    • Confidence intervals
    • Hypothesis testing, size, power
    • Uniformly most powerful tests
    • Likelihood ratio tests

Markov Processes, Queues and Simulation core material:

  • Simulation:
    • Inverse transform
    • Acceptance-rejection
  • 
Markov processes and queues:
    • Markov property
    • Homogeneous process
    • Irreducibility
    • Stationary distributions
    • Detailed-balance condition
    • Limit behavior
    • Time-reversibility
    • Probability transition matrix
    • Kolmogorov-Chapman equation
    • Recurrence and transience
    • Periodicity
    • Positive and null recurrence
    • First-step method
    • Rate matrix
    • Forward and backward equations
    • Exit and hitting distributions
    • Queues and queueing networks
  • Homogeneous Poisson processes:
    • Properties and characterizations
    • Thinning, superposition, and conditioning
    • Compound Poisson process