Random variables assign values to outcomes of random experiments. This document discusses: 1) The cumulative distribution function (CDF) defines the probability that a random variable is less than or equal to a value. CDFs for continuous and discrete variables are calculated differently. 2) Probability density functions (PDFs) and probability mass functions (PMFs) characterize the distributions of continuous and discrete random variables. 3) Key properties of random variables include expected value (mean), variance, and moments. Common distributions like normal, uniform, and exponential are introduced.

1. Chapter 2: Random Variables
Probability and Random Process (EEEg-2114)

2. 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
2

3. 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.


Line
al
Re
x
X 
)
(
x
A
B
3

4. 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:
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}.
}
TTT
,
,
,
,
,
,
,
{ TTH
HTT
THT
THH
HTH
HHT
HHH


0
1
1
1
2
2
2
3
:
)
(
:


X
TTT
TTH
HTT
THT
THH
HTH
HHT
HHH
4

5. The Cumulative Distribution Function
 The cumulative distribution function (cdf) of a random variable
X is defined as the probability of the event {X≤ x}.
Properties of the cdf, FX(x):
 The cdf has the following properties.
1
)
(
0
i.e.,
function,
negative
-
non
a
is
)
(
.

 x
F
x
F
i
X
X
1
)
(
lim
. 


x
F
ii X
x
0
)
(
lim
. 


x
F
iii X
x
)
(
)
( x
X
P
x
FX 

5

6. The Cumulative Distribution Function Cont’d…..
)
(
)
(
then
,
If
i.e.,
,
of
function
decreasing
-
non
a
is
)
(
.
2
1
2
1 x
F
x
F
x
x
X
x
F
iv
X
X
X


)
(
)
(
)
(
. 1
2
2
1 x
F
x
F
x
X
x
P
v X
X 



)
(
1
)
(
. x
F
x
X
P
vi 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.
6

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.



















3
,
1
3
2
,
8
/
7
2
1
,
2
/
1
1
0
,
8
/
1
0
,
0
)
(
x
x
x
x
x
x
FX
7

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.,
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,……




 du
u
f
x
FX )
(
)
(
8

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.,
Properties of the pdf, fX(x):
dx
x
dF
x
f X
X
)
(
)
( 










2
1
)
(
)
(
.
1
)
(
.
0
)
(
,
of
values
all
For
.
2
1
x
x
X
X
X
dx
x
f
x
X
x
P
iii
dx
x
f
ii
x
f
X
i
9

10. The Probability Mass Function
 The probability mass function (pmf) of a discrete random
variable X is defined as:
Properties of the pmf, PX (xi ):
)
(
)
(
)
(
)
( 1




 i
X
i
X
i
X
i
X x
F
x
F
x
P
x
X
P
 






k
k
X
k
X
i
X
x
P
iii
k
x
x
x
P
ii
k
x
P
i
1
)
(
.
.....
2,
,
1
,
if
,
0
)
(
.
.....
2,
,
1
,
1
)
(
0
.
10

11. Calculating the Cumulative Distribution Function
 The cdf of a continuous random variable X can be obtained by
integrating the pdf, i.e.,
 Similarly, the cdf of a discrete random variable X can be obtained by
using the formula:



x
X
X du
u
f
x
F )
(
)
(




x
x
k
k
X
X
k
x
x
U
x
P
x
F )
(
)
(
)
(
11

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:
 Similarly, the expected value of a discrete random variable X is
given by:
 Mean represents the average value of the random variable in a
very large number of trials.





 dx
x
xf
X
E X
X )
(
)
(




k
k
X
k
X x
P
x
X
E )
(
)
(

12

13. Expected Value, Variance and Moments Cont’d…..
ii. Variance
 The variance of a continuous random variable X, denoted by
σ2
X or VAR(X), is defined as:
 Expanding (x-μX )2 in the above equation and simplifying the
resulting equation, we will get:











dx
x
f
x
X
Var
X
E
X
Var
X
X
X
X
X
)
(
)
(
)
(
]
)
[(
)
(
2
2
2
2




2
2
2
)]
(
[
)
(
)
( X
E
X
E
X
Var
X 



13

14. Expected Value, Variance and Moments Cont’d…..
 The variance of a discrete random variable X is given by:
 The standard deviation of a random variable X, denoted by σX, is
simply the square root of the variance, i.e.,
iii.Moments
 The nth moment of a continuous random variable X is defined as:
 


k
k
X
X
k
X x
P
x
X
Var )
(
)
(
)
( 2
2


)
(
)
( 2
X
Var
X
E X
X 

 






 1
,
)
(
)
( n
dx
x
f
x
X
E X
n
n
14

15. Expected Value, Variance and Moments Cont’d…..
 Similarly, the nth moment of a discrete random variable X is
given by:
 Mean of X is the first moment of the random variable X.
1
,
)
(
)
( 
  n
x
P
x
X
E
k
k
X
n
k
15

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:
 The corresponding distribution function is given by:
where
.
2
1
)
(
2
2
2
/
)
(
2





 x
X e
x
f
 








 


x
y
X
x
G
dy
e
x
F




 2
2
2
/
)
(
2
2
1
)
(
dy
e
x
G y
x
2
/
2
2
1
)
( 





16

17. Some Special Distributions Cont’d……
 The normal or Gaussian distribution is the most common
continuous probability distribution.
)
(x
fX
x

Fig. Normal or Gaussian Distribution
17

18. Some Special Distributions Cont’d……
2. Uniform Distribution
3. Exponential Distribution
)
(x
fX
x
a b
a
b 
1
Fig. Uniform Distribution








otherwise.
0,
,
0
,
1
)
(
/
x
e
x
f
x
X


)
(x
fX
x
Fig. Exponential Distribution









otherwise.
0,
,
1
)
(
b
x
a
a
b
x
fX
18

19. Some Special Distributions Cont’d……
4. Gamma Distribution
5. Beta Distribution
where










otherwise.
0,
,
0
,
)
(
)
(
/
1
x
e
x
x
f
x
X
















otherwise.
0,
,
1
0
,
)
1
(
)
,
(
1
)
(
1
1
x
x
x
b
a
x
f
b
a
X 





1
0
1
1
.
)
1
(
)
,
( du
u
u
b
a b
a

19

20. Some Special Distributions Cont’d……
6. Rayleigh Distribution
7. Cauchy Distribution
8. Laplace Distribution








otherwise.
0,
,
0
,
)
(
2
2
2
/
2
x
e
x
x
f
x
X


.
,
)
(
/
)
( 2
2







 x
x
x
fX




.
,
2
1
)
( /
|
|





 
x
e
x
f x
X


20

21. Some Special Distributions Cont’d….
i. Discrete Probability Distributions
1. Bernoulli Distribution
2. Binomial Distribution
3. Poisson Distribution
.
)
1
(
,
)
0
( p
X
P
q
X
P 



.
,
,
2
,
1
,
0
,
)
( n
k
q
p
k
n
k
X
P k
n
k











 
.
,
,
2
,
1
,
0
,
!
)
( 


 

k
k
e
k
X
P
k


21

22. Some Special Distributions Cont’d….
4. Hypergeometric Distribution
5. Geometric Distribution
6. Negative Binomial Distribution
, max(0, ) min( , )
( )
m N m
k n k
N
n
m n N k m n
P X k
 
 
 
 
   
   
 
 
 
 


   
 
.
1
,
,
,
2
,
1
,
0
,
)
( p
q
k
pq
k
X
P k





 
1
( ) , , 1, .
1
r k r
k
P X k p q k r r
r


 
   
 

 
22

23. Random Variable Examples
Example-1:
The pdf of a continuous random variable is given by:
.
of
variance
and
mean
the
Evaluate
.
)
1
4
/
1
(
Find
.
.
of
cdf
ing
correspond
the
Find
.
.
of
value
the
Determine
.
constant.
a
is
re
whe
otherwise
,
0
1
0
,
)
(
X
d
X
P
c
X
b
k
a
k
x
kx
x
fX






 


23

24. Random Variable Examples Cont’d……
Solution:
2
1
2
1
0
1
2
1
1
)
(
.
2
1
0
















 




k
k
x
k
kxdx
dx
x
f
a X
otherwise
,
0
1
0
,
2
)
(


 



x
x
x
fX
24

25. Random Variable Examples Cont’d……
Solution:
0
2
)
(
)
(
1
0
for
:
2
Case
0
for
,
0
)
(
since
,
0
)
(
0
for
:
1
Case
)
(
)
(
:
by
given
is
of
cdf
The
.
2
0 0
2
X
x
x
u
udu
du
u
f
x
F
x
x
x
f
x
F
x
du
u
f
x
F
X
b
x x
X
X
X
x
X
X











 
 

25

26. Random Variable Examples Cont’d……
Solution:
1
,
1
1
0
,
0
,
0
)
(
by
given
is
cdf
The
1
0
1
2
)
(
)
(
1
for
:
3
Case
2
1
0
1
0
2
















 
x
x
x
x
x
F
u
udu
du
u
f
x
F
x
X
X
X
26

27. Random Variable Examples Cont’d……
Solution:
16
/
15
)
1
4
/
1
(
16
/
15
)
4
/
1
(
1
)
1
4
/
1
(
)
4
/
1
(
)
1
(
)
1
4
/
1
(
cdf
the
Using
.
16
/
15
)
1
4
/
1
(
16
/
15
4
/
1
1
)
1
4
/
1
(
2
)
(
)
1
4
/
1
(
pdf
the
Using
.
)
1
4
/
1
(
.
2
2
1
4
/
1
1
4
/
1





























 
X
P
X
P
F
F
X
P
ii
X
P
x
X
P
dx
x
dx
x
f
X
P
i
X
P
c
X
X
X
27

28. Random Variable Examples Cont’d……
Solution:
18
/
1
)
3
/
2
(
2
/
1
)
(
2
/
1
2
)
(
)
(
)]
(
[
)
(
)
(
Variance
.
3
/
2
0
1
3
2
2
)
(
)
(
Mean
.
Variance
and
Mean
.
2
2
1
0
1
0
3
2
2
2
2
2
3
1
0
1
0
2

















 
 
x
Var
dx
x
dx
x
f
x
X
E
X
E
X
E
X
Var
ii
x
dx
x
dx
x
xf
X
E
i
d
X
X
X
X
X
X




28

29. Random Variable Examples Cont’d……..
Example-2:
Consider a discrete random variable X whose pmf is given by:
.
of
variance
and
mean
the
Find
otherwise
,
0
1
0,
,
1
,
3
/
1
)
(
X
x
x
P
k
k
X




 


29

30. Random Variable Examples Cont’d……
Solution:
3
/
2
)
0
(
3
/
2
)
(
3
/
2
]
)
1
(
)
0
(
)
1
[(
3
/
1
)
(
)
(
)]
(
[
)
(
)
(
Variance
.
0
)
1
0
1
(
3
/
1
)
(
)
(
Mean
.
2
2
1
1
2
2
2
2
2
2
2
2
1
1



























x
Var
x
P
x
X
E
X
E
X
E
X
Var
ii
x
P
x
X
E
i
X
k
k
X
k
X
k
k
X
k
X



30

31. 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:
 We can determine the cdf and pdf of Y in terms of that of X.
 Consider some of the following functions.
31
)
(X
g
Y 
)
(X
g
Y 
b
aX 
2
X
|
| X
X
)
(
|
| x
U
X
X
e
X
log
X
sin
X
1

32. 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.
3. Obtain fY(y) from FY(y) by using direct differentiation, i.e.,
32
)
(
)
)
(
(
)
( y
Y
P
y
X
g
P
y
FY 



dy
y
dF
y
f Y
Y
)
(
)
( 

33. 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:
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
33
)]
(
[
)
(
)
(
)
( y
h
f
dy
y
dh
x
f
dy
dx
y
f X
X
Y 


34. Functions of a Random Variable Cont’d…..
ii. Determine the derivative of function g(xi ) at every real root xi ,
i.e. ,
iii. Find the pdf of Y by using the following formula.
34

 

i
i
X
i
i
X
i
i
Y x
f
x
g
x
f
dy
dx
y
f )
(
)
(
)
(
)
(
dy
dx
x
g i
i 
)
(

35. Examples on Functions of One Random Variable
Examples:
35
.
of
pdf
the
determine
,
tan
If
].
2
,
2
[
interval
in the
uniform
is
X
variable
random
The
.
).
(
Find
.
1
Let
.
).
(
Find
.
Let
.
).
(
Find
.
Let
.
2
Y
X
Y
d
y
f
X
Y
c
y
f
X
Y
b
y
f
b
aX
Y
a
Y
Y
Y









36. Examples on Functions of One Random Variable…..
Solutions:
36
)
(
1
)
(
)
(
)
(
)
(
)
(
)
(
0
that
Suppose
Method
Using
.
.
i
a
b
y
f
a
dy
y
dF
y
f
a
b
y
F
y
F
a
b
y
X
P
y
b
aX
P
y
Y
P
y
F
a
I
i
b
aX
Y
a
X
Y
Y
X
Y
y





 








 






 












37. Examples on Functions of One Random Variable…..
Solutions:
37
)
(
1
)
(
)
(
1
)
(
)
(
)
(
)
(
then
,
0
if
other
On the
Method
Using
.
.
ii
a
b
y
f
a
dy
y
dF
y
f
a
b
y
F
y
F
a
b
y
X
P
y
b
aX
P
y
Y
P
y
F
a
I
i
b
aX
Y
a
X
Y
Y
X
Y
y





 









 







 












38. Examples on Functions of One Random Variable…..
Solutions:
38
a
a
b
y
f
a
y
f
ii
i
I
i
b
aX
Y
a
X
Y all
for
,
1
)
(
:
obtain
we
,
)
(
and
)
(
equations
From
Method
Using
.
.





 





39. Examples on Functions of One Random Variable…..
Solutions:
39
    




 
















a
b
y
f
a
y
f
y
h
f
dy
y
dh
x
f
dy
dx
y
f
a
dy
dx
a
dy
y
dh
dy
dx
y
h
a
b
y
x
y
IR
Y
b
aX
Y
II
ii
b
aX
Y
a
X
Y
X
X
Y
1
)
(
)
(
)
(
)
(
1
1
)
(
solution
principal
the
is
)
(
,
any
For
is
of
space
range
the
and
one
-
to
-
one
is
function
The
Method
Using
.
.

40. Examples on Functions of One Random Variable…..
Solutions:
40
y
and
by
given
solutions
two
are
there
,
0
each
For
0
is
of
space
range
the
and
one
-
to
-
one
not
is
function
The
.
2
1
2






x
y
x
y
y
Y
X
Y
b

41. Examples on Functions of One Random Variable…..
Solutions:
41
   
 






















otherwise
y
y
f
y
f
y
y
f
x
f
dy
dx
x
f
dy
dx
y
f
x
f
dy
dx
y
f
y
dy
dx
y
dy
dx
y
dy
dx
y
dy
dx
b
X
X
Y
X
X
Y
i
X
i
i
Y
,
0
0
,
2
1
)
(
)
(
)
(
)
(
)
(
)
(
2
1
2
1
and
2
1
2
1
.
2
2
1
1
2
2
1
1

42. Examples on Functions of One Random Variable…..
Solutions:
42
 
   
 
0
/
,
y
1
1
)
(
1
1
)
(
)
(
)
(
)
(
1
)
(
solution
principal
the
is
)
(
1
,
any
For
0
/
is
of
space
range
the
and
one
-
to
-
one
is
1
function
The
.
2
2
2
IR
f
y
y
f
y
f
y
y
f
y
h
f
dy
y
dh
x
f
dy
dx
y
f
y
dy
y
dh
dy
dx
y
h
y
x
y
IR
Y
X
Y
c
X
Y
X
Y
X
X
Y





























43. Examples on Functions of One Random Variable…..
Solutions:
43
   






















y
y
y
f
y
y
f
y
h
f
dy
y
dh
x
f
dy
dx
y
f
y
dy
y
dh
dy
dx
y
h
y
x
y
Y
X
Y
d
Y
Y
X
X
Y
,
)
1
(
1
)
(
1
/
1
)
(
)
(
)
(
)
(
1
1
)
(
solution
principal
the
is
)
(
tan
,
any
For
)
,
(
is
of
space
range
the
and
one
-
to
-
one
is
tan
function
The
.
2
2
2
1



44. Examples on Functions of One Random Variable…..
Solutions:
44

45. Assignment-II
1. The continuous random variable X has the pdf given by:
.
of
variance
and
mean
the
.
)
1
(
.
.
of
cdf
the
.
.
of
value
the
.
:
Find
constant.
a
is
re
whe
otherwise
,
0
2
0
,
)
2
(
)
(
2
X
d
X
P
c
X
b
k
a
k
x
x
x
k
x
fX





 



45

46. Assignment-II Cont’d…..
2. The cdf of continuous random variable X is given by:
.
of
variance
and
mean
the
.
.
of
pdf
the
.
.
of
value
the
.
:
Determine
constant.
a
is
re
whe
1
1,
1
0
,
0
,
0
)
(
X
c
X
b
k
a
k
x
x
x
k
x
x
X
F












46

47. 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.
47