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Chapter 5
Introduction to Random Variables
Consider a random experiment described by a triplet (
S,
F
,P
). In applica
tions of probabilities, we are often interested in numerical quantities derived
from the experimental outcomes. These quantities may be viewed as func
tions from the sample space
S
into the set of real numbers
R
, as in:
s
∈
S
→
X
(
s
)
∈
R
Provided certain basic requirements are satisﬁed, these quantities are gener
ally called
random variables
.
As an example, consider the sum of the two numbers showing up when rolling
a fair die twice:
•
The set of all possible outcomes is
S
=
{
(
i,j
) :
∈ {
1
,
2
,...,
6
}}
.
•
The sum of the two numbers showing up may be represented by the
functional relationship
s
= (
)
→
X
(
s
) =
i
+
j.
•
Note that the function
X
(
s
) may be used in turn to deﬁne more complex
events. For instance, the event that the sum is greater or equal to 11
may be expressed concisely as
A
=
{
s
∈
S
:
X
(
s
)
≥
11
}
117
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The terminology
random variable
is appropriate in this type of situations
because:
•
The value
X
(
s
) depends on the experimental outcome
s
.
•
The outcome
s
of a particular realization of the random experiment
(i.e. a trial) is unknown beforehand, and so is
X
(
s
).
•
Each experimental trial may lead to a diﬀerent value of
X
(
s
)
Random variables are extremely important in engineering applications. They
are often used to model physical quantities of interest that cannot be pre
dicted exactly due to uncertainties. Some examples include:
•
Voltage and current measurements in an electronic circuit.
•
Number of erroneous bits per second in a digital transmission.
•
Instantaneous background noise amplitude at the output of an audio
ampliﬁer.
Modelization of such quantities as random variables allows the use of proba
bility in the design and analysis of these systems.
This and the next few Chapters are devoted to the study of random variables,
including: deﬁnition, characterization, standard models, properties, and a lot
more.
..
In this Chapter, we give a formal deﬁnition of a random variable, we introduce
the concept of a cumulative distribution function and we introduce the basic
types of random variables.
c
±
2003 Benoˆ
ıt Champagne
Compiled February 10, 2012
5.1 Preliminary notions
119
5.1 Preliminary notions
Function from
S
into
R
:
•
Let
S
denote a sample space of interest.
•
A function from
S
into
R
is a mapping, say
X
, that associate to every
outcome in
S
a unique real number
X
(
s
):
S
mapping X
real axis
s
1
s
2
s
3
X(s
1
)
X(s
2
)=X(s
3
)
Figure 5.1: Illustration of a mapping
X
from
S
into
R
.
•
The following notation is often used to convey this idea:
X
:
s
∈
S
→
X
(
s
)
∈
R
.
(5.1)
•
We refer to the sample space
S
as the
domain
of the function
X
.
•
The
range
of the function
X
, denoted
R
X
, is deﬁned as
R
X
=
{
X
(
s
) :
s
∈
S
} ⊆
R
(5.2)
That is,
R
X
is the of all possible values for
X
(
s
), or equivalently, the
set of all real numbers that can be “reached” by the mapping
X
.
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This note was uploaded on 02/12/2012 for the course ECSE 305 taught by Professor Champagne during the Spring '09 term at McGill.
 Spring '09
 Champagne

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