Week 1: Preliminaries I: Introduction to probability, review of distributions: Bernoulli, Binomial, Poisson, Multinomial, Exponential, Gamma, Gaussian distribution
Week 2: Preliminaries II: Conditional probability, Conditional expectation and variance, Computations with conditioning, Central limit theorem, Software demonstration of simulating discrete and continuous random variables
Week 3: DTMC I: Discrete time stochastic processes, Discrete time Markov chains, transition probabilities, Chapman-Kolmogorov equations, classification of states, Software demonstration of the concepts
Week 4: DTMC II: Limiting probabilities, Connection to Perron Frobenius Theorem, Mean time spent in transient states, Branching processes, Time reversible Markov chains, Applications
Week 5: Discrete time counting processes: Bernoulli random processes, definitions and alternate synthesis approaches via interarrivals, properties, operating on Bernoulli processes like merging and splitting, Applications to simple discrete-time queues
Week 6: Continuous time counting processes: Poisson processes, Interarrival and waiting time distributions, merging and splitting operations, order statistics, conditional distribution on arrival times, marked and compound Poisson processes, Applications
Week 7: CTMC I: Birth and Death processes, Transition probability function.
Week 8: CTMC II : Kolmogorov’s backward and forward equations, Limiting probabilities, Applications
Week 9: Renewal : definitions, examples, Limit theorems, renewal reward, Applications to reliability
Week 10: Applications to Queuing theory: basic definitions of queues and Kendall notation, analysis of M/M/X/X queues
Week 11: Martingale and Brownian motion: definition, connections to other processes, Hitting times, Gambler’s ruin problem, Brownian motion with drift, Geometric Brownian motion, White noise, Gaussian processes, Stationary and weak stationary processes
Week 12: Applications: Option pricing, risk neutral pricing, Arbitrage theorem, Black Scholes option pricing formula
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