LECTURE 1 0F 7
11.1 RANDOM VARIABLES
11.2 DISCRETE RANDOM
VARIABLES
(a) define random variable
(b) identify discrete and continuous
variables
(c) Understand probability
distribution functions
A random variable (RV)
~is a function that assigns a numerical
value to each simple event in a sample
space.
~ RV will be denoted by uppercase letters X
or Y and its values are denoted by lowercase
letters x or y
Example :
If two coins are tossed and a letter
X is used to represent the number of
heads, then x = 0, 1, 2.
Example :
If two dice are rolled and a letter
Y is used to represent the sum of
the number shown on dice,
then, y = 2, 3, 4,…12
A random variable can be discrete or
continuous
A Discrete Random Variable
~ is a function with countable or exact values
Example:
• the number of cars sold during a given
month
•
the number of houses in a certain block
A Continuous Random Variable
~ is a function with values contained in one or
more intervals
Example:
•
the height of a person
•
the time taken to complete an exam
Probability Distribution of a Discrete
Random Variable
~ list all the possible values that the random
variables can assume and their corresponding
probabilities.
If X is a discrete random variable then:
0  P  X= xi   1
k
and
 PX  x   1
i 1
i
The probability distribution can be
presented in the form of
i) table
ii) function
iii) graph
Example 1:
If two fair coins are tossed then the
possible outcomes are:
{HH,HT,TH,TT}
Let X represents the number of tail
obtained.
So,
x = 0,1,2
Probability distribution for X
x
(i) table
P(X = x)
(ii) function
1

4
P(X  x)  
1

2
0
1
2
1
4
1
2
1
4
(iii) graph
x  0, 2
P(X=x)
1
_
2
x 1
1
_
4
x
Example 2:
A fair die is rolled. If X represent the number
on die, show that X is a discrete random
variable. Find P(X < 3) and P 1< X  5 .
Solution:
X = the number on die
The probability distribution of X
x
P(X=x)
6
 PX
i 1
1
2
3
4
5
6
1
6
1
6
1
6
1
6
1
6
1
6
 xi   1
Thus, X is a discrete random variable.
x
P(X=x)
1
2
3
4
5
6
1
6
1
6
1
6
1
6
1
6
1
6
P  X < 3 
P(X=1)+ P  X = 2
1
=

6
2 1


6 3
1
6
x
P(X=x)
1
2
3
4
5
6
1
6
1
6
1
6
1
6
1
6
1
6
P 1< X  5 
= P  X = 2  P  X = 3 +P  X = 4 +P  X =5
4
=
6
2
=
3
Example 3
Random variable X has the following
probability distribution
x  1, 2,3, 4
kx

P  X  x   k 10  x  x  5, 6, 7
0
otherwise

where k is a constant
a) Determine the value of k
b) Construct a probability distribution table
c) Find (i) P  X>2
(ii) P  X= 4 or X = 6 
iii)P  X - 2  1
solution
(a)
x  1, 2,3, 4
kx

P  X  x   k 10  x  x  5, 6, 7
0
otherwise

k (1  2  3  4)  k (5  4  3)  1
22k  1
1
k
22
(b) The probability distribution table
x  1, 2,3, 4
kx

P  X  x   k 10  x  x  5, 6, 7
0
otherwise

1
k
22
x
1
2
3
4
5
6
7
P  X  x
1
22
2
22
3
22
4
22
5
22
4
22
3
22
1
2
3
4
5
6
7
PX  x 1
2
22
3
22
4
22
5
22
4
22
3
22
x
22
(c) (i) P  X > 2  1  P( X  1)  P( X  2)
1
2
19
 1


22 22
22
4
4
(ii) P  X = 4 or X = 6 

22 22
8
4


22 11
(iii) P  X-2  1  P  -1< X - 2 < 1
= P 1 < X < 3
= P  X = 2
2
=
22
1

11
CONCLUSION
X is known as a discrete random
variable if:
P  X = xi   0
k
 PX
i 1
 xi   1
0  P  X=x   1