ABSTRACT
A quick tour of probabilistic tools and techniques to study nondeterministic events is provided in this chapter. Initially axioms of probability theory are stated. The concept of random variable is next introduced. This is followed by a description of average measures such as expectation of a random variable. Typical second order measures, and the concept of independent random variables are also introduced. Common tools for studying distributions are z-transforms and moment generating functions. These ideas are further clarified via examples of discrete and continuous random variables. The well-known multivariate Gaussian distribution is also defined. Some well-known results like: Bienaymé-Chebyshev inequality, Chernoff bounds, Jensen’s inequality, weak and strong law of large numbers, Gaussian and Lévy’s central limit theorems, and stable distributions are also described. Distribution of range of a sequence of random variables is also studied. This is particularly useful in modeling Internet traffic. Elements of the theory of large deviation are outlined in the last section.
