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 ...
