27: Introduction to Stochastic Processes

"Science is based on uncertainty" - Lewis Thomas."While writing my book (Stochastic Processes, 1953) I had an argument with Feller. He asserted that everyone said 'random variable' and I asserted that everyone said 'chance variable.' We obviously had to use the same name in our books, so we decided the issue by a stochastic procedure. That is, we tossed for it and he won." - Joe Doob."It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge" – Laplace.

Lecture outline: Probability theory and Stochastic theory basics

1. What is a stochastic process?

Deterministic vs. stochastic process

What is a random variable (RV)?

Discrete RVs: Binomial, Poisson, Geometric RVs

Continuous RVs: Normal; Exponential; Gamma; Erlang; Weibull; Log-normal; Pareto

Probability models of Continuous RVs:

Cumulative distribution function (CDF) and Probability density function (PDF)

Expectation of a random variable

What is a stochastic process? (family on indexed random variables)

2. Time/state based classification

Discrete-time Discrete-state stochastic process;

Discrete-time Continuous-state stochastic process;

Continuous-time Discrete-state stochastic process;

Continuous -time Continuous-state stochastic process

3. Representing stochastic processes

Multiple ways of seeing the same dynamic system

Arrival process; Interarrival process; Counting process

Primary reference for this lecture:

“High Speed Networks and Internets: Performance and Quality of Service” by William Stallings; Chapter 7: “Overview of Probability and Stochastic Processes”

Secondary references for this lecture:

1. “Fundamentals of Performance Evaluation of Computer and Telecommunication System”, by Obaidat and Boudriga; Chapter 2: “Probability Theory Review”.

2. “Cartoon Guide to Statistics” by Larry Gonick; Chapter 3: “Probability” and Chapter 4: “Random Variables”.