Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 36

function (distribution) of the noise process. Typically for a random noise signal,
S ¼ R (the set of real numbers).

1.3.2.2 Random Variables

A random variable X is a real-valued function whose domain is the set of all
possible events (i.e., the sample space S) of a repeating experiment.

X : S ! M ; where M R ð M is a subset of R Þ

Example In a coin-tossing experiment, if one defines X(h) = 1, X(t) =-1, then
X:{h, t}: ? {1, -1} is a random variable.

Notes:

1. Random variables can be discrete (as in the above example) or continuous (as
in the amplitude of Gaussian noise).
2. In case of noise where S ¼

R , one can define the random variable X as the noise

amplitude itself. That is:

X : R ! R j X ð r Þ¼ r 8 r 2 R

1.3.2.3 Joint Probability

Assume that one has two experiments, Experiment 1 and Experiment 2. Assume
also that A is a possible outcome (or event) for Experiment 1 and that B is a
possible outcome for Experiment 2. Then the joint probability of A and B (denoted
by p(A \ B)) is the probability that A is the outcome of Experiment 1 and B the
outcome of Experiment 2.

Example If n(t) is noise, and one defines the events A = {n(t 1 ) [ 0.1} and
B = {n(t 2 ) [ 0.5}, then

p ð A \ B Þ¼ p f n ð t 1 Þ [ 0 : 1 and n ð t 2 Þ [ 0 : 5 g:

that is, p(A \ B) is the probability that the sample of noise at t 1 is greater than 0.1
and the sample of noise at t 2 is greater than 0.5.

1.3.2.4 Conditional Probability

Conditional probability (CP) is the probability of an event A given that an event
B has already occurred. CP is denoted by the expression p(A|B). The conditional
probability and the joint probability are related to one another according to: