Appendix A

PROBABILITY AND STATISTICS OVERVIEW

**A.1 PROBABILITY THEORY**

Defining a sample space (outcomes), Ω, a field (events), *B,* and a probability function (on a class of events), Pr, we can construct an *experiment* as the triple, {Ω, *B,* Pr}.

**Example A.l**

Consider the experiment, {Ω, *B,* Pr} of tossing a fair coin, then we see that

Sample space: Ω = {*H,T*}

Events: *B* = {0, {*H*}, {*T*}}

Probability: Pr(*H*) = *p*

Pr(*T*) = 1 −p

With the idea of a sample space, probability function, and experiment in mind, we can now start to define the concept of a discrete random signal more precisely. We define a discrete *random variable* as a real function whose value is determined by the outcome of an experiment. It assigns a real number to each point of a sample space Ω, which consists of all the possible outcomes of the experiment. A random variable *X* and its realization *x* are written as

Consider the following example of a simple experiment.

**Example A.2**

We are asked to analyze the experiment of flipping a fair coin, then the sample space consists of a head or tail as possible outcomes, that is,

If we assign a 1 for a head and 0 for a tail, then the random variable *X* performs the mapping of

where x(.) is called the sample value or realization of the random variable ...