5-discrete dis-1.ppt

Chapter 5
Discrete
Distributions
Prepared By: Janvi Joshi 5-1

Discrete vs Continuous Distributions
• Random Variable -- a variable which contains the
outcomes of a chance experiment
• Discrete Random Variable -- the set of all possible
values is at most a finite or a countable infinite
number of possible values.
• This will produce value that are nonnegative whole
numbers
– Randomly selecting 25 people who consume soft
drinks & determining how many people prefer diet soft
drinks
– Determining the number of defects in a batch of 50
items
– Counting the number of people who arrive at store
during a five minute period.

• Continuous Random Variable -- takes on values at
every point over a given interval.
– It is normally generated from time, height, weight &
volume.
• Elapsed time between arrivals of bank customers
• Percent of the labor force that is unemployed

Some Special Distributions
• Discrete
– Binomial
– Poisson
– hypergeometric
• Continuous
– normal
– uniform
– exponential
– t
– chi-square
– F

Requirements for a
Discrete Probability Function
• Probabilities are between 0 and 1,
inclusively
• Total of all probabilities equals 1
P X
( )
over all x
  1
X
all
for
1
)
(
0 
 X
P

Requirements for a Discrete
Probability Function -- Examples
X P(X)
-1
0
1
2
3
.1
.2
.4
.2
.1
1.0
X P(X)
-1
0
1
2
3
-.1
.3
.4
.3
.1
1.0
X P(X)
-1
0
1
2
3
.1
.3
.4
.3
.1
1.2
VALID NOT
VALID
NOT
VALID

µ = ∑ (Xi * P(Xi))
where µ is the long run average,
Xi = the ith outcome
• Var(Xi) = ∑ ((Xi – m)2* P(Xi))
• Standard Deviation is computed by taking the
square root of the variance

Discrete Distribution -- Example
0
1
2
3
4
5
0.37
0.31
0.18
0.09
0.04
0.01
Number of
Crises
Probability
Distribution of Daily
Crises
0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5
P
r
o
b
a
b
i
l
i
t
y
Number of Crises
Find out mean, variance & Standard deviation for given example

Mean of the Crises Data Example
 
m    

E X X P X
( ) .
115
0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5
P
r
o
b
a
b
i
l
i
t
y
Number of Crises
X P(X) XP(X)
0 .37 .00
1 .31 .31
2 .18 .36
3 .09 .27
4 .04 .16
5 .01 .05
1.15

Binomial Distribution
• A binomial experiment exhibits the following
four properties
1. The experiment consists of a sequence of n
identical trials
2. Two outcome possible on each trials. one as
success & the other as a failure
3. The probability of success , denoted by p & not
change from trial to trial. The probability of a
failure denoted by 1-p
4. Trials are independent.

Binomial Distribution
• Probability
function
• Mean
value
• Variance and
standard
deviation
 
P X
n
X n X
X n
X n X
p q
( )
!
! !


  

for 0
m  
n p
2
2

 
  
   
n p q
n p q

According to the U.S. Census Bureau, approximately
6% of all workers in Jackson, are unemployed. In
conducting a random telephone survey in Jackson,
what is the probability of getting two or fewer
unemployed workers in a sample of 20?find out
variance & SD for same.
Binomial Distribution:
Demonstration Problem 5.3
5-13
Prepared By: Janvi Joshi

Binomial Distribution:
Demonstration Problem 5.3
• In this example,
– 6% are unemployed => p
– The sample size is 20 => n
– 94% are employed => q
– X is the number of successes desired
– What is the probability of getting 2 or fewer
unemployed workers in the sample of 20? =>
P(X≤2)
5-14

Binomial Distribution: Demonstration
Problem 5.3
n
p
q
P X P X P X P X



      
   
20
06
94
2 0 1 2
2901 3703 2246 8850
.
.
( ) ( ) ( ) ( )
. . . .
   
P X
( )
)!
( )( )(. ) .
. .
 

 

0
20!
0!(20 0
1 1 2901 2901
0 20 0
06 94
   
P X
( )
!( )!
( )(. )(. ) .
. .
 

 

1
20!
1 20 1
20 06 3086 3703
1 20 1
06 94
   
P X
( )
!( )!
( )(. )(. ) .
. .
 

 

2
20!
2 20 2
190 0036 3283 2246
2 20 2
06 94

n = 20 PROBABILITY
X 0.1 0.2 0.3 0.4
0 0.122 0.012 0.001 0.000
1 0.270 0.058 0.007 0.000
2 0.285 0.137 0.028 0.003
3 0.190 0.205 0.072 0.012
4 0.090 0.218 0.130 0.035
5 0.032 0.175 0.179 0.075
6 0.009 0.109 0.192 0.124
7 0.002 0.055 0.164 0.166
8 0.000 0.022 0.114 0.180
9 0.000 0.007 0.065 0.160
10 0.000 0.002 0.031 0.117
11 0.000 0.000 0.012 0.071
12 0.000 0.000 0.004 0.035
13 0.000 0.000 0.001 0.015
14 0.000 0.000 0.000 0.005
15 0.000 0.000 0.000 0.001
16 0.000 0.000 0.000 0.000
17 0.000 0.000 0.000 0.000
18 0.000 0.000 0.000 0.000
19 0.000 0.000 0.000 0.000
20 0.000 0.000 0.000 0.000
   
n
p
P X C


  
20
40
10 01171
20 10
10 10
40 60
.
( ) .
. .

Ex:5.5
Ex:5.6

Ex: 5.9 What is the first big change that American
drivers made due to higher gas prices? According to
an Access America survey, 30% said that it was
cutting recreational driving. However, 27% said that
it was consolidating or reducing errands. If these
figures are true for all American drivers, and if 15
such drivers are randomly sampled and asked what
is the first big change they made due to higher gas
prices,
a. What is the probability that exactly 6 said that it
was consolidating or reducing errands?
b. What is the probability that none of them said
that it was cutting recreational driving?
c. what is the probability that more than 9 said that
it was cutting recreational driving?
5-21

Poisson Distribution
• Describes discrete occurrences over a
continuum or interval
• It is normally used to to describe the
number of random arrival per some time
interval.
• Each occurrence is independent any other
occurrences.
• The number of occurrences in each interval
can vary from zero to infinity.
• Ex: Binomial dis: To determine prob of
number of US made cars for repair if take
20 samples.
• Ex: Poisson distribution: A number of cars
randomly arriving for repair facility during
10min intervals.

Poisson Distribution
• Probability function
P X
X
X
where
long run average
e
X
e
( )
!
, , , ,...
:
. ...
 
 





for
(the base of natural logarithms )
0 1 2 3
2 718282

• Suppose bank customers arrive randomly
on weekday afternoons at an average of 3.2
customers every 4 mins. What is the
probability of exactly 5 customers arriving
in a 4- minute interval on a weekday
afternoon?

Poisson Distribution: Demonstration
Problem 5.8
0528
.
0
!
10
=
)
10
=
(
!
=
P(X)
minutes
8
customers/
4
.
6
=
Adjusted
minutes
8
customers/
10
=
X
minutes
4
customers/
2
.
3
4
.
6
10
X
6.4 



e
e
X
P
X






Poisson Approximation
of the Binomial Distribution
• Binomial probabilities are difficult to calculate
when n is large.
• Under certain conditions binomial
probabilities may be approximated by Poisson
probabilities.
• Poisson approximation
If and the approximation is acceptable .
n n p
  
20 7,
Use   
n p.

• n = 50 and p = .03. What is the probability
that x = 4? That is, P(x = 4, n = 50 and p =
.03) = ?

Hypergeometric Distribution
• Sampling without replacement from a finite
population
• The number of objects in the population is
denoted N.
• Each trial has exactly two possible outcomes,
success and failure.
• Trials are not independent
• X is the number of successes in the n trials
• The binomial is an acceptable approximation, if
n < 5% N. Otherwise it is not.

Hypergeometric Distribution
• Probability function
– N is population size
– n is sample size
– A is number of successes in population
– x is number of successes in sample
  
P x
C C
C
A x N A n x
N n
( ) 
 

• Twenty-four people, of whom eight are women,
apply for a job. If five of the applicants are
sampled randomly, what is the probability that
exactly three of those sampled are women?

• Catalog Age lists the top 17 U.S. firms in annual catalog
sales. Dell Computer is number one followed by IBM
and W.W. Grainger. Of the 17 firms on the list, 8 are in
some type of computer-related business. Suppose five
firms are randomly selected.
a. What is the probability that none of the firms is in
some type of computer-related business?
b. What is the probability that all five firms are in some
type of computer-related business?
c. What is the probability that exactly three are in non-
computer-related business?

• A western city has 19 police officers eligible for
promotion. Ten of the 19 are Hispanic. Suppose only
five of the police officers are chosen for promotion
and that one is Hispanic. If the officers chosen for
promotion had been selected by chance alone, what
is the probability that one or fewer of the five
promoted officers would have been Hispanic?

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