Text: G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes Second Edition, Oxford University, 1993

1. Review of Basic Probability

Probability Spaces - Independence Random - Variables Distribution Function - Random Vectors Conditional - Distribution Expectation and its Properties - Conditional Expectation - Characteristic function - Generating function

2. Sequences of Random Variables

Modes of Convergence - Relationships between convergence modes - Borel Cantelli Lemmas - Weak Law of Large Numbers - Strong Law of Large Numbers for independent sequences - Central Limit Theorem: Lyapunov and Lindeberg-Feller Theorems - Large Deviations for iid sequences - Martingales

3. Random Processes

Continuous and discrete-time processes - Stationarity - Wide-Sense Stationarity - Ergodicity - Mean Ergodic Theorem - Bikhoff-Khinchin Ergodic Theorem - Spectral Density of wide sense stationary processes - WSS processes and linear-time-invariant systems - Linear Minimum Mean-Square Estimation and Filtering - Karhunen-Loeve Expansion of Continuous-Time Processes

4. Renewal Processes

Memoryless property of exponential distribution - Poisson Processes - Nonhomogeneous Poisson Processes - Laws of large numbers for renewal processes; renewal theorem - Stopping Rules and Wald's identity - Renewal theory: steady-state distribution of age and residual life.

5. Markov Processes

(Discrete-Time) Markov Chains - Homogenous Markov chains - Irreducible chains - Positive Recurrent, Null Recurrent, Transient States and Mean recurrence time - Periodicity and ergodic states - Stationary distributions and limit theorems - Reversibility and Detailed Balance Equations - Continuous-Time Markov processes - Transition rate matrix and Holding times - Stationary distributions - Reversibility - Birth-Death Markov Processes - Applications to Queueing theory: exponential service queues