Elementary Probability Theory - Digital Communication: Principles and System Modelling - page 207

number of outcome is infinite/uncountable. For those cases, the sample space is
defined as continuous.

A.1.2 Events

An event is a subset A of the sample space S, i.e. it is a set of possible outcomes.
If the outcome of an experiment is an element of A we say that the event A has
occurred. An event consisting of a single point of S is often called a simple or
elementary event .
As an example, let's take the random experiment of tossing a coin twice. Here,
two events are defined as A and B, where A be the event that 'at least one head
occurs' and V be the event that 'the second toss results in a tail.' Then A = {HT,

TH, HH}, V = {HT, TT},

A.1.3 Probability-Understanding Approaches

From the study of random experiment, it is clear that, in any random experiment
there is always uncertainty as to whether a particular event will or will not occur. As
a measure of the chance or probability with which we can expect the event to occur
it is convenient to assign a number between 0 and 1. If we are sure or certain that
the event will occur we say that its probability is 100% or 1, but if we are sure that
the event will not occur we say that its probability is zero.

Definition approach 1 If an event can occur in h different ways out of a total number
of n possible ways, all of which are equally likely, then the probability of the event
is h / n [1].
Definition approach 2 If after n ( n is a large number) repetitions of an experiment,
an event is observed to occur in h of these, then the probability of the event is h / n .
This is also called the empirical probability of the event.
Mathematically, probability of a favorable event h from a sample space S is
defined as

h
n

=

p ( h ) Lt (A.1)

n →∞

where, n is the total number of chance experiments.

A.2 Random Variable

Random variable is neither random nor variable [2]; conversely, it can be defined
as a function of the elements of a sample space S . Generally random variable is
represented by a capital letter (X) and any particular value of the random variable
by a lower case letter (x). Thus, given an experiment defined by a sample space S