Page 92 - Fundamentals of Communications Systems

3.6 Chapter Three

3.2 Random Variables
In communications, the information transmission is often corrupted by random
noise. This random noise manifests itself as a random voltage (a real number)
at the output of the electrical circuits in the receiver. The concept of a random
variable (RV) links the axiomatic definition of probability with these observed
real random numbers. A random variable has an underlying random exper-
iment. The set of all experimental outcomes are denoted and a particular
outcome of the random experiment is denoted with ω.
Definition 3.6 Let ( , F, P ) be a probability space. A real random variable X(ω)is a
single-valued function or mapping from to the real line (R).
There are three types of random variables: discrete, continuous, and mixed.
A discrete random variable has a finite (or countably infinite) number of possible
values. A continuous random variable takes values in some interval of the real
line of nonzero length. A mixed random variable is a convex combination of a
discrete and a continuous random variable. To simplify the notation when no
ambiguity exists, X represents the random variable X(ω) (the experimental
outcome index is dropped) and x = X(ω) represents a particular realization of
this random variable.

Definition 3.7 An observed real number resulting from the random experiment is
denoted a sample from the random variable.

EXAMPLE 3.8
Matlab has a built-in random number generator. In essence this function when executed
performs an experiment and produces a real number output (a sample of the random
variable). Each time this function is run, a different real value is returned. Go to Matlab
and type rand(1) and see what happens. Each time you run this function, it returns a
number that is unable to be predicted. The output can be characterized in many ways,
but the outputs are not completely predictable.

Random variables are completely characterized by either of two related func-
tions: the cumulative distribution function (CDF) or the probability density
1
function (PDF) . These functions are the subject of the next two sections.

3.2.1 Cumulative Distribution Function
Definition 3.8 For a random variable X(ω), the CDF is a function F X (x) defined as

F X (x) = P ({ω : X(ω) ≤ x}) ∀ x ∈ R

1 Continuous RVs have PDFs but discrete random variables have probability mass functions
(PMF).