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 2 S ! X(s) 2 R
Provided certain basic requirements are satis▯ed, 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 = f(i;j) : i;j 2 f1;2;:::;6gg.
▯ The sum of the two numbers showing up may be represented by the
s = (i;j) ! X(s) = i + j:
▯ Note that the function X(s) may be used in turn to de▯ne more complex
events. For instance, the event that the sum is greater or equal to 11
may be expressed concisely as A = fs 2 S : X(s) ▯ 11g
The terminology random variable is appropriate in this type of situations
▯ 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 di▯erent 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
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: de▯nition, characterization, standard models, properties, and a lot
In this Chapter, we give a formal de▯nition of a random variable, we introduce
the concept of a cumulative distribution function and we introduce the basic
types of random variables.
2003 Beno^▯t 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):
s2 real axis
Figure 5.1: Illustration of a mapping X from S into R.
▯ The following notation is often used to convey this idea:
X : s 2 S ! X(s) 2 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 , iX de▯ned as
R X fX(s) : s 2 Sg ▯ 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.
2003 Beno^▯t Champagne Compiled February 10, 2012 5.1 Preliminary notions 120
▯ Let X be a function from S into R.
▯ We de▯ne the inverse function X as follows: for any subset D of R,
X (D) = fs 2 S : X(s) 2 Dg (5.3)
▯ That is, X (D) is the subset of S containing all the outcomes s (possibly
more than one) such that X(s) is in D. This is illustrated below.
mapping X -1
X (D) D=[a,b]
Figure 5.2: Illustration of inverse mapping X
▯ This de▯nition of an inverse is very general; it applies even in the case
when X is many-to-one.
Left semi-in▯nite intervals:
▯ To every x 2 R, we associate a left semi-in▯nite interval I x de▯ned as
Ix, (▯1;x] = fy 2 R : y ▯ xg (5.4)
▯ The inverse image of I xnder mapping X is given by
X ▯1 (Ix) = fs 2 S : X(s) ▯ xg (5.5)
2003 Beno^▯t Champagne Compiled February 10, 2012 5.2 Definition of a random variable 121
5.2 De▯nition of a random variable
De▯nition: Let (S;F;P) be a probability space. A function X : s 2 S !
X(s) 2 R is called a random variable (RV) if
X ▯1(I ) = fs 2 S : X(s) ▯ xg 2 F; for all x 2 R (5.6)
▯ According to this de▯nition, X de▯nes a mapping from sample space S
into R, as illustrated in Figure 5.1
▯ However, X is not arbitrary: we require that for any x 2 R, the inverse
image X (Ix), as illustrated in Figure 5.3, be a valid event.
mapping X -1
Ix real axis
X (I ) x
Figure 5.3: Inverse image X(I )
▯ This condition ensures that P(fs 2 S : X(s) ▯ xg), i.e. the probability
that X(s) belong to the interval I , xs well-de▯ned.
2003 Beno^▯t Champagne Compiled February 10, 2012 5.2 Definition of a random variable 122
I A fair coin is
ipped twice. Let random variable X represent the number of tails
observed in this experiment. Here, the sample space may be de▯ned as
S = fHH;HT;TH;TTg
Since this is a ▯nite set, a proper choice of event algebra is
F = P S f;;fHHg;:::;Sg
Note that F contains 2 = 16 events. According to the problem statement, the
function X : S ! R may be computed as follows:
s = HH ! X(s) = 0
s = HT or TH ! X(s) = 1
s = TT ! X(s) = 2
so that its range isXR= f0;1;2g. This is illustrated below:
For any x 2 R, we have
X (x ) = fs 2 S : X(s) ▯ xg 2 F
since x is a subset of S and F = S contains all the possible subsets of S. This
shows that X is a valid random variable. For instance:
x < 0 ) X ▯1(x ) = ; 2 F
0 ▯ x < 1 ) X (x ) = fHHg 2 F
1 ▯ x < 2 ) X (x ) = fHH;HT;THg 2 F
2 ▯ x ) X (x ) = S 2 F
2003 Beno^▯t Champagne Compiled February 10, 2012 5.2 Definition of a random variable 123
Remarks on condition (5.6):
▯ By de▯nition, X ▯1(I ) = fs 2 S : X(s) ▯ xg is a subset of S. Therefore,
whenever F = P , cSndition (5.6) is satis▯ed and we need not worry
about it. This is the case when S is ▯nite or countably in▯nite.
▯ When S is uncountably in▯nite and F = B , theSe exist functions X :
S ! R that do not satisfy condition (5.6) and for which P(fs 2 S :
X(s) ▯ xg) is not de▯ned. It is precisely to avoid this situation that
(5.6) is included in the de▯nition of a random variable.
▯ In applications, we will want to compute probabilities of the type
P(fs 2 S : X(s) 2 Dg)
where D ▯ R represents any practical subset of real numbers. This
includes, for example, intervals of the type [a;b], for any a ▯ b 2 R, as
well as unions, intersections and/or complements of such intervals.
▯ As a consequence of condition (5.6), it can be shown that
fs 2 S : X(s) 2 Dg) 2 F
for any D 2 B ,Rso that P(fs 2 S : X(s) 2 Dg) is well-de▯ned for any
practical subset of real-numbers (see next page for more on this).
▯ In this course, we shall work with relatively simple functions X and shall
always assume that the condition X (Ix) 2 F is satis▯ed.
2003 Beno^▯t Champagne Compiled February 10, 2012 5.2 Definition of a random variable 124
Further remarks (optional reading):
▯ The condition X (Ix) 2 F in the de▯nition of random variable X ensures that
X (x ) = fs 2 S : X(s) ▯ xg
is a valid event, for which the probability P(X(Ix)) is well-de▯ned.
▯ More importantly, the condition ensures that for practical subsets of real numbers
encountered in applications of probability, i.e. for any D 2RB , the Borel ▯eld of R,
the set of experimental outcomes
X (D) = fs 2 S : X(s) 2 Dg
is also a valid event for which a probability can be computed.
▯ While the detail of the proof are beyond the scope of this course, the justi▯cation of
this statement involves three basic steps:
- Any real number subset D 2 B R can be expressed as a combination of unions,
intersections and complements of basic intervals of the typexI .
- Because X (Ix) 2 F for any x and because F is a ▯-algebra (closed under union,
intersection and complement), it follows that X (D) is also in F.
- Finally, X (D) 2 F implies that P(X (D)) is well-de▯ned.
▯ In the next section, we will see how P(X (Ix)) can actually be used in the computation
of P(X (D)).
2003 Beno^ ▯t Champagne Compiled February 10, 2012 5.