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