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ROHANA
ROHANABINTI
BINTIABDUL
ABDULHAMID
HAMID
INSTITUT
INSTITUTE FOR
E FORENGINEERING
ENGINEERINGMATHEMATICS
MATHEMATICS(IMK)
(IMK)
UNIVERSITIMALAYSIA
MALAYSIAPERLIS
PERLIS
UNIVERSITI
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CHAPTER 3
PROBABILITY DISTRIBUTION
(PART 1)
PROBABILITY DISTRIBUTION
• 3.1
Introduction
Binomial
distribution
• 3.2
• 3.3
Poisson
distribution
Normal
distribution
• 3.4
3.1 INTRODUCTION

A probability distribution is obtained
when probability values are assigned to
all possible numerical values of a
random variable.

Probability distribution can be classified
either discrete or continuous.
• BINOMIAL
DISTRIBUTION
DISCRETE
DISTRIBUTIONS • POISSON DISTRIBUTION
CONTINUOS • NORMAL DISTRIBUTION
DISTRIBUTIONS
3.2 THE BINOMIAL DISTRIBUTION
Definition 3.1 :
An experiment in which satisfied the following
characteristic is called a binomial experiment:
1. The random experiment consists of n identical
trials.
2. Each trial can result in one of two outcomes,
which we denote by success, S or failure, F.
3. The trials are independent.
4. The probability of success is constant from trial to
trial, we denote the probability of success by p and
the probability of failure is equal to (1 - p) = q.
Definition 3.2 :
A binomial experiment consist of n identical
trial with probability of success, p in each
trial. The probability of x success in n
trials
is given by
P( X  x)  Cx p q
n

x = 0, 1, 2, ......, n
x
n x
Definition 3.3 :The Mean and Variance of X
If X ~ B(n,p), then
Mean
Variance
where
 n is the total number of trials,
 p is the probability of success and
 q is the probability of failure.
Standard
deviation
EXAMPLE 3.1
Given that X ~ b(12, 0.4), find
a) P ( X  2)
b) P ( X  3)
c) P ( X  4)
d) P (2  X  5)
e) E( X )
f) Var( X )
SOLUTIONS
a) P ( X  2) 
12
C2 (0.4) 2 (0.6)10
 0.0639
b) P ( X  3) 
12
C3 (0.4)3 (0.6)9
 0.1419
c) P ( X  4) 
12
C4 (0.4) 4 (0.6)8
 0.2128
d) P (2  X  5)  P ( X  2)  P ( X  3)  P ( X  4)
 0.0639  0.1419  0.2128
=0.4185
e) E ( X )  np
= 12(0.4)
=4.8
f) Var ( X )  npq
= 12(0.4)(0.6)
= 2.88
Exercise
• In Kuala Lumpur, 30% of workers take
public transportation daily. In a sample of
10 workers,
I. What is the probability that exactly three
workers take public transportation daily?
II.What is the probability that at least three
workers take public transportation daily?
III.Calculate the standard deviation of this
distribution.
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Page 12
3.3 The Poisson Distribution
Definition 3.4
A random variable X has a Poisson
distribution and it is referred to as a
Poisson random variable if and only if its
probability distribution is given by

e   x
P( X  x) 
for x  0,1, 2,3,...
x!


λ (Greek lambda) is the long run mean
number of events for the specific time or space
dimension of interest.
A random variable X having
distribution can also be written as
X ~ Po ( )
with E ( X )   and Var ( X )  
a
Poisson
EXAMPLE 3.2
Given that X ~ Po (4.8), find
a) P( X  0)
b) P( X  9)
c) P( X  1)
SOLUTIONS
a) P ( X  0) 
b) P( X  9) 
e
4.8
0
4.8
9
e
4.8
 0.0082
0!
4.8
 0.0307
9!
c) 1  P ( X  0)  1  0.0082
= 0.9918
EXAMPLE 3.3
Suppose that the number of errors in a piece of
software has a Poisson distribution with
parameter   3 . Find
a) the probability that a piece of software has no
errors.
b) the probability that there are three or more
errors in piece of
software .
c) the mean and variance in the number of errors.
SOLUTIONS
e 3  30
a) P( X  0) 
0!
 e3  0.050
b)P( X  3)  1  P( X  0)  P( X  1)  P( X  2)
e 3  30 e 3  31 e 3  32
 1


0!
1!
2!
3 9
3  1
 1 e    
1 1 1 
 1  0.423  0.577
Exercise 1
• Phone calls arrive at the rate of 48 per
hour at the reservation desk for Regional
Airways
I. Find the probability of receiving three calls
in a 5-minutes interval time.
II.Find the probability of receiving more than
two calls in 15 minutes.
Powerpoint Templates
Page 19
Exercise 2
• An average of 15 aircraft accidents occurs
each year. Find
I. The mean, variance and standard
deviation of aircraft accident per month.
II.The probability of no accident during a
months.
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Page 20
IMPORTANT!!!!
exactly two
=2
More than two/
Exceed two
2
Two or more/
At least two/
Two or more
2
less than two/
Fewer than two
At most two/
Two or fewer/
Not more than
two
2
2