Download Random Variables and Probability Distribution (1)

Random Variables and
Probability Distribution (1)
• Definition: A random variable is a
function that associates a real number
with each element in the sample space.
• A random variable symbol should be
capital letter such X, Y, … and its
corresponding is small letter.
Random Variables and
Probability Distribution (1)
• Example: Two balls are drawn in
succession without replacement from
an urn containing 4 red balls and 3
black balls. The possible outcomes and
the value Y of the random variable Y
where Y is the number of red balls are:
• Sample space “S”: [RR, RB, BR, BB]
• Number of reds “Y”: [2, 1, 1, 0]
Random Variables and
Probability Distribution (1)
• Definition:
• If a sample space contains a finite
number of possibilities or an unending
sequence with as many elements as
there are whole numbers, it is called a
discrete sample space.
Random Variables and
Probability Distribution (1)
• Definition:
• If a sample space contains an infinite
number of possibilities equal to the
number of points on a line segment, it
is called a continuous sample space.
Random Variables and
Probability Distribution (1)
• Example: Classify the following random
variables are discrete or continuous:
• X: The number of automobile accidents?
• It is discrete random variable.
• Y: The length of time to play 18 holes of
Golf?
• It is continuous random variable.
• M: The mount of milk produced yearly by a
particular cow?
• It is continuous random variable.
Random Variables and
Probability Distribution (1)
• Example:
• N: The number of eggs laid each month by a
hen?
• It is discrete random variable.
• P: The number of building permits issued
each monthly in a certain city?
• It is discrete random variable.
• Q: The weight of grain produced per acre?
• It is continuous random variable.
Random Variables and
Probability Distribution (1)
Discrete Probability Distribution
• Definition :
• The set of ordered pairs (x, f(x)) is a
probability function, probability mass
function or probability distribution of
the discrete random variable X if for
each possible outcome x ;
Random Variables and
Probability Distribution (1)
As
f ( x)  0
 f ( x)  1
x
P( X  x)  f ( x)
Random Variables and
Probability Distribution (1)
• Example (1):
• A shipment of 20 similar laptop
computers to a retail outlet contains 3
that are defective. If a school makes a
random purchase of 2 of these
computers,
find
the
probability
distribution for the number
of
defectives.
Random Variables and
Probability Distribution (1)
• Solution:
• Let X be a random variable whose
values x are the possible numbers of
defective computers purchased by the
school. Then x can only take the
numbers 0, 1, 2 then:
 3 17 

0

2 

3!17!
136
68



 
f (0)  P ( X  0) 


(3  0)!0!15!2!
190
95
 20 


2 
20!


2!18!
Random Variables and
Probability Distribution (1)
2.
 3 17 
  
1 1 
3!17!
3  17 51

f (1)  P( X  1) 



2!16!
19
190
 20 
 
20  19  18!
2 
18!2!
 3 17 
  
2  0 
3

f (2)  P( X  2) 

190
 20 
 
2 
Random Variables and
Probability Distribution (1)
Then the probability distribution is
X
f(x)
0
1
2
68/95 51/190 3/190
68 51
3
2  68  51  3
1
x f ( x)  95  190  190 
190
Random Variables and
Probability Distribution (1)
• Example2:
• Suppose X is the sum of up faces of 2
dies:
# of ways 2 die can sum to x
f ( x)  P ( X  x) 
# of 2 die can result in
f ( x)  P( X  x)  Pr obabilitym mass function
Random Variables and
Probability Distribution (1)
• X: sum of up faces of 2 dies
1
2
3
4
5
6
1
2
2
3
3
4
4
5
5
6
6
7
3
4
5
4
5
6
5
6
7
6
7
8
7
8
9
8
9
10
6
7
7
8
8
9
9
10
10
11
11
12
Random Variables and Probability
Distribution (1)
• The probability
distribution function
table is
X
2
f(x) = P (X=x)
1/36
3
4
5
2/36
3/36
4/36
6
7
8
9
5/36
6/36
5/36
4/36
10
11
12
3/36
2/36
1/36
Random Variables and
Probability Distribution (1)
• Then
1  2  3  4  5  6  5  4  3  2  1 36

1
x f ( x) 
36
36