Appendix A

Elements of Probability Theory

In this appendix, we report basic notions of probability theory and stochastic process theory. The material of this appendix is based on Feller (1957), on Appendix B.1 of Santi (2005), and on online resources such as Wikipedia and Wolfram MathWorld.

A *sample space* Ω is a set representing all possible outcomes of a certain random experiment. A sample space is *discrete* if it is composed of a finite number of elements (e.g., outcomes of a coin toss experiment), or of infinitely many elements that can be arranged into a simple sequence *e*_{1}, *e*_{2}, ….

A *random variable X* is a function defined on a sample space. If the sample space on which *X* is defined is discrete, *X* is said to be a discrete random variable.

Examples of random variables are the number of heads in a sequence of *k* coin tosses (discrete random variable), the position of a certain particle in a physical system, the position of a node moving according to a certain mobility model at a certain instant of time, and so on.

Let *X* be a discrete random variable, and let Ω = {*x*_{1}, *x*_{2}, …, *x*_{j}, …} be the set of possible values of *X*. The function

is called the *probability distribution* (also called *probability mass function*) of the random variable ...

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