Addis Ababa Institute of Technology(AAiT)
Department of Electrical & Computer Engineering
Probability and Random Process (EEEg-2114)
Chapter 2: Random Variables
Random Variables
Outline






Introduction
The Cumulative Distribution Function
Probability Density and Mass Functions
Expected Value, Variance and Moments
Some Special Distributions
Functions of One Random Variable
Semester-I, 2017
By Habib M.
2
Introduction
 A random variable X is a function that assigns a real number
X(ω) to each outcome ω in the sample space Ω of a random
experiment.
 The sample space Ω is the domain of the random variable and the
set RX of all values taken on by X is the range of the random
variable.
 Thus, RX is the subset of all real numbers.

X ( )  x
x
Semester-I, 2017

A
B
By Habib M.
Re al Line
3
Introduction Cont’d……
 If X is a random variable, then {ω: X(ω)≤ x}={X≤ x} is an event
for every X in RX.
Example: Consider a random experiment of tossing a fair coin three
times. The sequence of heads and tails is noted and the sample
space Ω is given by:
  {HHH , HHT , HTH , THH , THT , HTT , TTH , TTT }
Let X be the number of heads in three coin tosses. X assigns
each possible outcome ω in the sample space Ω a number from
the set RX={0, 1, 2, 3}.
 : HHH HHT HTH THH THT HTT TTH TTT
X ( ) : 3
Semester-I, 2017
2
2
2
By Habib M.
1
1
1
0
4
The Cumulative Distribution Function
 The cumulative distribution function (cdf) of a random variable
X is defined as the probability of the event {X≤ x}.
FX ( x)  P( X  x)
Properties of the cdf, FX(x):
 The cdf has the following properties.
i. FX ( x) is a non - negative function, i.e.,
0  FX ( x)  1
ii. lim FX ( x)  1
x 
iii. lim FX ( x)  0
x 
Semester-I, 2017
By Habib M.
5
The Cumulative Distribution Function Cont’d…..
iv. FX ( x) is a non - decreasing function of X , i.e.,
If x1  x2 , then FX ( x1 )  FX ( x2 )
v. P( x1  X  x2 )  FX ( x2 )  FX ( x1 )
vi. P( X  x)  1  FX ( x)
Example:
Find the cdf of the random variable X which is defined as the
number of heads in three tosses of a fair coin.
Semester-I, 2017
By Habib M.
6
Semester-I, 2017
By Habib M.
7
The Cumulative Distribution Function
Solution:
 We know that X takes on only the values 0, 1, 2 and 3 with
probabilities 1/8, 3/8, 3/8 and 1/8 respectively.
 Thus, FX(x) is simply the sum of the probabilities of the
outcomes from the set {0, 1, 2, 3} that are less than or equal to x.
0, x  0
1 / 8, 0  x  1

 FX ( x)  1 / 2, 1  x  2
7 / 8, 2  x  3

1, x  3
Semester-I, 2017
By Habib M.
8
Types of Random Variables
 There are two basic types of random variables.
i. Continuous Random Variable

A continuous random variable is defined as a random variable
whose cdf, FX(x), is continuous every where and can be written as
an integral of some non-negative function f(x), i.e.,
FX ( x)  


f (u )du
ii. Discrete Random Variable

A discrete random variable is defined as a random variable whose
cdf, FX(x), is a right continuous, staircase function of X with
jumps at a countable set of points x0, x1, x2,……
Semester-I, 2017
By Habib M.
9
The Probability Density Function
 The probability density function (pdf) of a continuous random
variable X is defined as the derivative of the cdf, FX(x), i.e.,
dFX ( x)
f X ( x) 
dx
Properties of the pdf, fX(x):
i. For all values of X , f X ( x)  0
ii.



f X ( x)dx  1
x2
iii. P( x1  X  x2 )   f X ( x)dx
x1
Semester-I, 2017
By Habib M.
10
The Probability Mass Function
 The probability mass function (pmf) of a discrete random
variable X is defined as:
PX ( X  xi )  PX ( xi )  FX ( xi )  FX ( xi 1 )
Properties of the pmf, PX (xi ):
i. 0  PX ( xi )  1, k  1, 2, .....
ii. PX ( x)  0, if x  xk , k  1, 2, .....
iii.
P
X
( xk )  1
k
Semester-I, 2017
By Habib M.
11
Calculating the Cumulative Distribution Function
 The cdf of a continuous random variable X can be obtained by
integrating the pdf, i.e.,
FX ( x)  
x

f X (u )du
 Similarly, the cdf of a discrete random variable X can be obtained by
using the formula:
FX ( x) 
Semester-I, 2017
P
xk  x
X
( xk )U ( x  xk )
By Habib M.
12
Expected Value, Variance and Moments
i.
Expected Value (Mean)

The expected value (mean) of a continuous random variable X,
denoted by μX or E(X), is defined as:

 X  E ( X )   xf X ( x)dx


Similarly, the expected value of a discrete random variable X is
given by:
 X  E ( X )   xk PX ( xk )
k

Mean represents the average value of the random variable in a
very large number of trials.
Semester-I, 2017
By Habib M.
13
Expected Value, Variance and Moments Cont’d…..
ii. Variance

The variance of a continuous random variable X, denoted by
σ2X or VAR(X), is defined as:
 2 X  Var ( X )  E[( X   X ) 2 ]


2

X
 Var ( X )   ( x   X ) 2 f X ( x)dx

Expanding (x-μX )2 in the above equation and simplifying the
resulting equation, we will get:

2
Semester-I, 2017
X
 Var ( X )  E ( X )  [ E ( X )]
2
By Habib M.
2
14
Expected Value, Variance and Moments Cont’d…..
 The variance of a discrete random variable X is given by:
 2 X  Var( X )   ( xk   X ) 2 PX ( xk )
k
 The standard deviation of a random variable X, denoted by σX, is
simply the square root of the variance, i.e.,
 X  E ( X   X ) 2  Var( X )
iii.Moments
 The nth moment of a continuous random variable X is defined as:

E ( X )   x n f X ( x)dx ,
n

Semester-I, 2017
By Habib M.
n 1
15
Expected Value, Variance and Moments Cont’d…..
 Similarly, the nth moment of a discrete random variable X is
given by:
E ( X )   xk PX ( xk ) ,
n
n 1
k
 Mean of X is the first moment of the random variable X.
Semester-I, 2017
By Habib M.
16
Some Special Distributions
i.
Continuous Probability Distributions
1. Normal (Gaussian) Distribution

The random variable X is said to be normal or Gaussian
random variable if its pdf is given by:
1
f X ( x) 

e
2 2
 ( x   ) 2 / 2 2
.
The corresponding distribution function is given by:
x
1

2
FX ( x)  
2
e
 ( y   ) 2 / 2 2
 x 
dy  G

  
x
1 y /2
where G ( x)  
e
dy

2
2
Semester-I, 2017
By Habib M.
17
Some Special Distributions Cont’d……
 The normal or Gaussian distribution is the most common
continuous probability distribution.
f X (x)
x

Fig. Normal or Gaussian Distribution
Semester-I, 2017
By Habib M.
18
Some Special Distributions Cont’d……
2. Uniform Distribution
 1

, a xb
f X ( x)   b  a

 0, otherwise.
1
ba
f X (x)
a
x
b
Fig. Uniform Distribution
3. Exponential Distribution
f X (x)
1

 e  x /  , x  0,
f X ( x)   

 0, otherwise.
Semester-I, 2017
By Habib M.
x
Fig. Exponential Distribution
19
Some Special Distributions Cont’d……
4. Gamma Distribution
 x 1
x / 

e
, x  0,

f X ( x )   ( ) 

0, otherwise.

5. Beta Distribution
 1
x a 1 (1  x) b 1 , 0  x  1,

f X ( x )    ( a, b)

0,
otherwise.

where
 ( a , b) 
Semester-I, 2017
1

0
u a 1 (1  u ) b 1 du.
By Habib M.
20
Some Special Distributions Cont’d……
6. Rayleigh Distribution
x  x 2 / 2 2

 2e
, x  0,
f X ( x )  

 0, otherwise.
7. Cauchy Distribution
f X ( x) 
 /
  (x  )
2
2
,    x  .
8. Laplace Distribution
1 |x|/ 
f X ( x) 
e
,    x  .
2
Semester-I, 2017
By Habib M.
21
Some Special Distributions Cont’d….
i.
Discrete Probability Distributions
1. Bernoulli Distribution
P ( X  0)  q,
P( X  1)  p.
2. Binomial Distribution
 n  k n k
P( X  k )  
,
k 
p q
 
k  0,1,2,  , n.
3. Poisson Distribution
P ( X  k )  e 
Semester-I, 2017
k
k!
, k  0,1,2, , .
By Habib M.
22
Some Special Distributions Cont’d….
4. Hypergeometric Distribution
P( X  k ) 
m
 
k 
 
 N m 


 n k 


,
N 
 
n 
 
max(0, m  n  N )  k  min( m, n )
5. Geometric Distribution
P( X  k )  pqk , k  0,1,2,, ,
q  1  p.
6. Negative Binomial Distribution
 k  1 r k  r
P( X  k )  
p q
,

 r 1
Semester-I, 2017
By Habib M.
k  r , r  1,
.
23
Random Variable Examples
Example-1:
The pdf of a continuous random variable is given by:
kx ,
f X ( x)  
0 ,
0  x 1
otherwise
whe re k is a constant.
a. Determine the value of k .
b. Find the corresponding cdf of X .
c. Find P (1 / 4  X  1)
d . Evaluate the mean and variance of X .
Semester-I, 2017
By Habib M.
24
Random Variable Examples Cont’d……
Solution:
a.



f X ( x ) dx  1 
1

0
kxdx  1
 x2
 k 
 2
k
 1
2
k  2
2 x,
 f X ( x)  
0,
Semester-I, 2017
1
  1
0
0  x 1
otherwise
By Habib M.
25
Random Variable Examples Cont’d……
Solution:
b.
The cdf of X is given by :
FX ( x) 
Case 1 :

x

f X (u ) du
for x  0
FX ( x)  0, since f X ( x)  0, for x  0
Case 2 :
for 0  x  1
FX ( x) 
Semester-I, 2017

x
0
f X (u ) du 
By Habib M.

x
0
2udu  u
2
x
0
 x2
26
Random Variable Examples Cont’d……
Solution:
Case 3 :
for x  1
FX ( x ) 
1

0
f X (u ) du 
1

0
2udu  u
2
1
0
1
 The cdf is given by
0,
 2
FX ( x )   x ,
1,

Semester-I, 2017
x0
0  x 1
x 1
By Habib M.
27
Random Variable Examples Cont’d……
Solution:
c.
P (1 / 4  X  1)
i. Using the pdf
1
1
P (1 / 4  X  1)  
1/ 4
 P (1 / 4  X  1)  x
2
f X ( x) dx   2 xdx
1/ 4
1
1/ 4
 15 / 16
 P (1 / 4  X  1)  15 / 16
ii. Using the cdf
P (1 / 4  X  1)  FX (1)  FX (1 / 4)
 P (1 / 4  X  1)  1  (1 / 4) 2  15 / 16
 P (1 / 4  X  1)  15 / 16
Semester-I, 2017
By Habib M.
28
Random Variable Examples Cont’d……
Solution:
d.
Mean and Variance
i. Mean
1
1
0
0
 X  E ( X )   xf X ( x)dx   2 x 2 dx
 X
2 x3 1

 2/3
3 0
ii. Variance
 X 2  Var ( X )  E ( X 2 )  [ E ( X )]2
1
1
E ( X )   x f X ( x ) dx   2 x 3 dx  1 / 2
2
2
0
0
  X  Var ( x )  1 / 2  ( 2 / 3) 2  1 / 18
2
Semester-I, 2017
By Habib M.
29
Random Variable Examples Cont’d……..
Example-2:
Consider a discrete random variable X whose pmf is given by:
1 / 3 , xk  1, 0, 1
PX ( xk )  
0 ,
otherwise
Find the mean and variance of X .
Semester-I, 2017
By Habib M.
30
Random Variable Examples Cont’d……
Solution:
i. Mean
 X  E( X ) 
1
x
k  1
k
PX ( xk )  1 / 3(1  0  1)  0
ii. Variance
 X 2  Var ( X )  E ( X 2 )  [ E ( X )]2
E( X ) 
2
1
2
2
2
x
P
(
x
)

1
/
3
[(

1
)

(
0
)

(
1
)
]  2/3
 k X k
2
k  1
  X  Var ( x)  2 / 3  (0) 2  2 / 3
2
Semester-I, 2017
By Habib M.
31
Class Work
1.
2.
3.
4.
5.
Semester-I, 2017
By Habib M.
32
Functions of One Random Variable
 Let X be a continuous random variable with pdf fX(x) and suppose
g(x) is a function of the random variable X defined as:
Y  g(X )
 We can determine the cdf and pdf of Y in terms of that of X.
 Consider some of the following functions.
aX  b
sin X
1
X
log X
Semester-I, 2017
X2
Y  g( X )
|X |
X
eX
| X | U ( x)
By Habib M.
33
Functions of a Random Variable Cont’d…..
 Steps to determine fY(y) from fX(x):
Method I:
1. Sketch the graph of Y=g(X) and determine the range space of Y.
2. Determine the cdf of Y using the following basic approach.
FY ( y)  P( g ( X )  y)  P(Y  y)
3. Obtain fY(y) from FY(y) by using direct differentiation, i.e.,
dFY ( y )
fY ( y ) 
dy
Semester-I, 2017
By Habib M.
34
Functions of a Random Variable Cont’d…..
Method II:
1. Sketch the graph of Y=g(X) and determine the range space of Y.
2. If Y=g(X) is one to one function and has an inverse
transformation x=g-1(y)=h(y), then the pdf of Y is given by:
dx
dh( y )
fY ( y ) 
f X ( x) 
f X [h( y )]
dy
dy
3. Obtain Y=g(x) is not one-to-one function, then the pdf of Y can
be obtained as follows.
i. Find the real roots of the function Y=g(x) and denote them by xi
Semester-I, 2017
By Habib M.
35
Functions of a Random Variable Cont’d…..
ii. Determine the derivative of function g(xi ) at every real root xi ,
i.e. ,
dxi
g ( xi ) 
dy
iii. Find the pdf of Y by using the following formula.
f Y ( y)  
i
Semester-I, 2017
dxi
f X ( xi )   g ( xi ) f X ( xi )
dy
i
By Habib M.
36
Examples on Functions of One Random Variable
Examples:
a. Let Y  aX  b. Find f Y ( y ).
b. Let Y  X 2 . Find f Y ( y ).
c. Let Y 
1
. Find f Y ( y ).
X
d . The random variable X is uniform in the interval [
 
, ].
2 2
If Y  tan X , determine the pdf of Y .
Semester-I, 2017
By Habib M.
37
Examples on Functions of One Random Variable…..
Solutions:
a. Y  aX  b
i. Using Method  I
Suppose that a  0
y b

Fy ( y )  P (Y  y )  P (aX  b  y )  P X 

a 

 y b
FY ( y )  FX 

 a 
 f Y ( y) 
Semester-I, 2017
dFY ( y ) 1
 y b
 fX 

dy
a
 a 
By Habib M.
(i )
38
Examples on Functions of One Random Variable…..
Solutions:
a. Y  aX  b
i. Using Method  I
On the other if a  0, then
y b

Fy ( y )  P(Y  y )  P(aX  b  y )  P X 

a 

 y b
FY ( y )  1  FX 

 a 
dFY ( y )
1  y b
 f Y ( y) 
  fX 

dy
a  a 
Semester-I, 2017
By Habib M.
(ii)
39
Examples on Functions of One Random Variable…..
Solutions:
a. Y  aX  b
i. Using Method  I
From equations (i ) and (ii) , we obtain :
f Y ( y) 
Semester-I, 2017
1
 y b
fX 
,
a
 a 
for all a
By Habib M.
40
Examples on Functions of One Random Variable…..
Solutions:
a. Y  aX  b
ii. Using Method  II
The function Y  aX  b is one - to - one and the range
space of Y is IR
y b
 h( y ) is the principal solution
a
dx dh( y ) 1
dx
1

 

dy
dy
a
dy
a
For any y, x 
dx
dh y 
1
 y b
f Y ( y) 
f X ( x) 
f X h( y )   f Y ( y ) 
fX 

dy
dy
a
 a 
Semester-I, 2017
By Habib M.
41
Examples on Functions of One Random Variable…..
Solutions:
b. The function Y  X 2 is not one - to - one and the range
space of Y is y  0
For each y  0, there are two solutions given by
x1   y and x 2 
Semester-I, 2017
y
By Habib M.
42
Examples on Functions of One Random Variable…..
Solutions:
b.
dx1
dx1
1
1



and
dy
dy
2 y
2 y
dx2
dx2
1
1



dy
dy
2 y
2 y
f Y ( y)  
i
dxi
dx1
dx2
f X ( xi )  f Y ( y ) 
f X ( x1 ) 
f X ( x2 )
dy
dy
dy
  y   f  y ,
 1
2 y f X
 f Y ( y)  


Semester-I, 2017
X
0,
By Habib M.
y0
otherwise
43
Examples on Functions of One Random Variable…..
Solutions:
c. The function Y 
1
is one - to - one and the range
X
space of Y is IR /0
1
For any y, x   h( y ) is the principal solution
y
dx dh( y )
1

 2
dy
dy
y
f Y ( y) 
1
dx
dh y 
1
f X ( x) 
f X h( y )   f Y ( y )  2 f X  
dy
dy
y
 y
 f Y ( y) 
Semester-I, 2017
1
1
 ,
f
X 
2

y
y
IR /0
By Habib M.
44
Examples on Functions of One Random Variable…..
Solutions:
d . The function Y  tan X is one - to - one and the range
space of Y is (, )
For any y, x  tan 1 y  h( y ) is the principal solution
dx dh( y )
1


dy
dy
1 y2
dx
dh y 
1/ 
f Y ( y) 
f X ( x) 
f X h( y )   f Y ( y ) 
dy
dy
1 y2
1
 f Y ( y) 
,
2
 (1  y )
Semester-I, 2017
  y  
By Habib M.
45
Examples on Functions of One Random Variable…..
Solutions:
Semester-I, 2017
By Habib M.
46
Assignment-II
1. The continuous random variable X has the pdf given by:
k (2 x  x 2 ) , 0  x  2
f X ( x)  
0 ,
otherwise
whe re k is a constant.
Find :
a. the value of k .
b. the cdf of X .
c. P ( X  1)
d . the mean and variance of X .
Semester-I, 2017
By Habib M.
47
Assignment-II Cont’d…..
2. The cdf of continuous random variable X is given by:
x0
0 ,


F ( x )  k x , 0  x  1
X

x 1

1,
whe re k is a constant.
Determine :
a. the value of k .
b. the pdf of X .
c. the mean and variance of X .
Semester-I, 2017
By Habib M.
48
Assignment-II Cont’d…..
3. The random variable X is uniform in the interval [0, 1]. Find
the pdf of the random variable Y if Y=-lnX.
Semester-I, 2017
By Habib M.
49