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Random Variables
In application of probabilities, we are often concerned with numerical values which are
random in nature. For example, we may consider the number of customers arriving at
service station at a particular interval of time or the transmission time of a message in a
communication system. These random quantities may be considered as real-valued
function on the sample space. Such a real-valued function is called real random variable
and plays an important role in describing random data. We shall introduce the concept of
random variables in the following sections.
Mathematical Preliminaries
Real-valued point function on a set
Recall that a real-valued function f : S  maps each element s  S , a unique element
f ( s )  . The set S is called the domain of f and the set R f  { f ( x) | x  S } is called
the range of f . Clearly R f  . The range and domain of f are shown in Fig, .
f ( s1 )
s1
f ( s2 )
s2
s3
s4
Domain of
f
f ( s3 )
f ( s1 )
Range of
f
Image and Inverse image
For a point s  S , the functional value f ( s)  is called the image of the point s. If
A  S , then the set of the images of elements of A is called the image of A and denoted
by f ( A). Thus
f ( A)  { f ( s) | s  A}
Clearly f ( A)  R f
Suppose   . The set {x | f ( x)  } is called the inverse image of  under f and is
denoted by f 1 ().
Example Suppose S  {H , T } and f : S  is defined by f ( H )  1 and f (T )  1.
Therefore,



R f  {1, 1} 
Image of H is 1 and that of T is -1.
For a subset of
say 1  (, 1.5],
f 1 (1 )  {s | f ( s)  1}  {H , T }.
For another subset B2  [5, 9], f 1 ( B2 )  .
Continuous function
A real function f :  is said to be continuous at a point a if and only if
(i)
f (a ) is defined
(ii)
lim f ( x)  lim f ( x)  f (a)
x a 
x a 
The function f :  is said to be right-continuous
at a point a if and only if
(iii)
f (a ) is defined
lim f ( x)  f (a)
x a 
We can similarly define a left-continuous function.
f ( x)
a
Function continuous at
xa
x
Random variable
A random variable associates the points in the sample space with real numbers.
Consider the probability space ( S , F , P) and function X : S  mapping the sample
space S into the real line. Let us define the probability of a subset B 
by
PX ({B})  P( X 1 ( B))  P({s | X ( s)  B}). Such a definition will be valid if
( X 1 ( B)) is a valid event. If S is a discrete sample space, ( X 1 ( B)) is always a valid
event, but the same may not be true if S is infinite. The concept of sigma algebra is
again necessary to overcome this difficulty. We also need the Borel sigma algebra B the sigma algebra defined on the real line.
The function X : S  called a random variable if the inverse image of all Borel sets
under X is an event. Thus, if X is a random variable, then
X 1 ()  {s | X ( s) }  F.
X 1
1
A  X ( B)
B
S
Figure Random Variable
Notations:
 Random variables are represented by
upper-case letters.
 Values of a random variable are
denoted by lower case letters
 X (s)  x means that x is the value of a
random variable X at the sample point
s.
 Usually, the argument s is omitted and
we simply write X  x.
(To be animated)
Remark
 S is the domain of X .
 The range of X , denoted by RX , is given by
RX  { X (s) | s  S}.
Clearly RX 

.
The above definition of the random variable requires that the mapping X is such that
X 1 () is a valid event in S. If S is a discrete sample space, this requirement is
met by any mapping X : S  . Thus any mapping defined on the discrete sample
space is a random variable.
Example 1: Consider the example of tossing a fair coin twice. The psample space is S={
HH,HT,TH,TT} and all four outcomes are equally likely. Then we can define a random
variable X as follows
Sample Point
HH
HT
TH
TT
Value of the
random
Variable X  x
0
1
2
3
Here RX  {0,1, 2,3}.
Example 2: Consider the sample space associated with the single toss of a fair die. The sample
space is given by S  {1,2,3,4,5,6}
If we define the random variable X that associates a real number equal to the number in the
face of the die, then
X  {1,2,3,4,5,6}
Probability Space induced by a Random Variable
The random variable X induces a probability measure PX on B defined by
PX ({B})  P( X 1 ( B))  P({s | X ( s )  B})
.
The probability measure PX satisfies the three axioms of probability:
Axiom 1
PX ( B)  P( X 1 ( B))  1
.
Axiom 2
PX ( )  P( X 1 ( ))  P( S )  1
Axiom 3
Suppose B1 , B2 ,.... are disjoint Borel sets. Then X 1 ( B1 ), X 1 ( B2 ),.... are distinct events in
F. Therefore,


PX ( Bi )  P( X 1 ( Bi )
i 1
i 1

  P( X 1 ( Bi ))
i 1

  PX ( Bi )
i 1
Thus the random variable X induces a probability space ( S , B, PX )
Probability Distribution Function
We have seen that the event B and {s | X ( s)  B} are equivalent and
PX ({B})  P({s | X (s)  B}). The underlying sample space is omitted in notation and we
simply write { X  B} and P ({ X  B}) in stead of {s | X ( s)  B} and P({s | X ( s)  B})
respectively.
Consider the Borel set (, x] where x represents any real number. The equivalent
event X 1 ((, x])  {s | X ( s)  x, s  S} is denoted as { X  x} The event { X  x} can
be taken as a representative event in studying the probability description of a random
variable X . Any other event can be represented in terms of this event. For example,
{ X  x}  { X  x}c ,{x1  X  x2 }  { X  x2 } \{ X  x1},
{ X  x} 
1 

{ X  x} \{ X  x  }

n 1 
n 

and so on.
The probability P({ X  x})  P({s | X ( s)  x, s  S}) is called the probability
distribution function ( also called the cumulative distribution function abbreviated as
CDF) of X and denoted by FX ( x). Thus (, x],
FX ( x)  P({ X  x})
Value of the random variable
FX ( x)
Random
variable
Eaxmple 3 Consider the random variable X in Example 1
We have
P({ X  x})
Value of the
random
Variable X  x
1
0
1
2
3
4
1
4
1
4
1
4
For x  0,
FX ( x)  P({ X  x})  0
For 0  x  1,
FX ( x)  P({ X  x})  P({ X  0}) 
1
4
For 1  x  2,
FX ( x)  P({ X  x})
 P({ X  0}  { X  1})
 P({ X  0})  P({ X  1})
1 1 1
 
4 4 2
For 2  x  3,

FX ( x)  P({ X  x})
 P({ X  0}  { X  1}  { X  2})
 P({ X  0})  P({ X  1})  P({ X  2})
1 1 1 3
  
4 4 4 4
For x  3,

FX ( x)  P({ X  x})
 P( S )
1
Properties of Distribution Function

0  FX ( x)  1
This follows from the fact that FX (x) is a probability and its value should lie between
0 and 1.
 FX (x)
is
a
non-decreasing
function
of
Thus,
X.
x1  x2 , then FX ( x1 )  FX ( x2 )
x1  x2
 { X ( s)  x1 }  { X ( s )  x2 }
 P{ X ( s)  x1 }  P{ X ( s )  x2 }
 FX ( x1 )  FX ( x2 )

FX (x) is right continuous
FX ( x  )  lim FX ( x  h)  FX ( x)
h 0
h 0
Because, lim FX ( x  h)  lim P{ X ( s)  x  h}
h 0
h 0
h 0
h 0
= P{X ( s)  x}
=FX ( x )

FX ()  0
Because, FX ()  P{s | X (s)  }  P( )  0
 FX ()  1
Because, FX ()  P{s | X (s)  }  P(S )  1
 P({x1  X  x2 })  FX ( x2 )  FX ( x1 )
We have
{ X  x2 }  { X  x1} {x1  X  x2 }
 P({ X  x2 })  P({ X  x1})  P({x1  X  x2 })
 P({x1  X  x2 })  P({ X  x2 })  P({ X  x1})  FX ( x2 )  FX ( x1 )

FX ( x  )=FX ( x)  P( X  x)
FX ( x  )  lim FX ( x  h)
h 0
h 0
 lim P{ X ( s )  x  h}
h 0
h 0
= P{ X ( s )  x}  P( X ( s )  x)
=FX ( x)  P ( X  x)
We can further establish the following results on probability of events on the real line:
P{x1  X  x2 }  FX ( x2 )  FX ( x1 )  P( X  x1 )
P({x1  X  x2 })  FX ( x2 )  FX ( x1 )  P( X  x1 )  P( X  x2 )
P({X  x})  P({x  X  })  1  FX ( x)
Thus we have seen that given FX ( x), -  x  , we can determine the probability of any
event involving values of the random variable X . Thus FX ( x) x  X is a complete
description of the random variable X .
Example 4 Consider the random variable X defined by
FX ( x)  0,
1
1
 x
8
4
 1,
Find
x  2
2 x 0
x0
a) P(X = 0)
b) P  X  0
c) P  X  2
d) P 1  X  1
Solution:
a) P( X  0)  FX (0 )  FX (0 )
1 3

4 4
b) P  X  0  FX (0)
 1
1
c) P  X  2  1  FX (2)
 1 1  0
d) P 1  X  1
 FX (1)  FX (1)
1 7
 1 
8 8
FX ( x)
1
x