## 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