O'Reilly logo

Bayesian Signal Processing: Classical, Modern and Particle Filtering Methods by James V. Candy

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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

(A.1)

img

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,

img

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 ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required