Download x - Fairfield

Slides by
John
Loucks
St. Edward’s
University
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
1
Chapter 5
Discrete Probability Distributions






Random Variables
Developing Discrete Probability Distributions
Expected Value and Variance
Binomial Probability
Distribution
Poisson Probability
Distribution
Hypergeometric Probability
Distribution
.40
.30
.20
.10
0
1
2
3
4
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
2
Random Variables
A random variable is a numerical description of the
outcome of an experiment.
A discrete random variable may assume either a
finite number of values or an infinite sequence of
values.
A continuous random variable may assume any
numerical value in an interval or collection of
intervals.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
3
Discrete Random Variable
with a Finite Number of Values

Example: JSL Appliances
Let x = number of TVs sold at the store in one day,
where x can take on 5 values (0, 1, 2, 3, 4)
We can count the TVs sold, and there is a finite
upper limit on the number that might be sold (which
is the number of TVs in stock).
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
4
Discrete Random Variable
with an Infinite Sequence of Values

Example: JSL Appliances
Let x = number of customers arriving in one day,
where x can take on the values 0, 1, 2, . . .
We can count the customers arriving, but there is
no finite upper limit on the number that might arrive.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
5
Random Variables
Question
Family
size
Random Variable x
x = Number of dependents
reported on tax return
Distance from x = Distance in miles from
home to the store site
home to store
Own dog
or cat
x = 1 if own no pet;
= 2 if own dog(s) only;
= 3 if own cat(s) only;
= 4 if own dog(s) and cat(s)
Type
Discrete
Continuous
Discrete
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
6
Discrete Probability Distributions
The probability distribution for a random variable
describes how probabilities are distributed over
the values of the random variable.
We can describe a discrete probability distribution
with a table, graph, or formula.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
7
Discrete Probability Distributions
Two types of discrete probability distributions will
be introduced.
First type: uses the rules of assigning probabilities
to experimental outcomes to determine probabilities
for each value of the random variable.
Second type: uses a special mathematical formula
to compute the probabilities for each value of the
random variable.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
8
Discrete Probability Distributions
The probability distribution is defined by a
probability function, denoted by f(x), that provides
the probability for each value of the random variable.
The required conditions for a discrete probability
function are:
f(x) > 0
f(x) = 1
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
9
Discrete Probability Distributions
There are three methods for assign probabilities to
random variables: classical method, subjective
method, and relative frequency method.
The use of the relative frequency method to develop
discrete probability distributions leads to what is
called an empirical discrete distribution.
example
on next
slide
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
10
Discrete Probability Distributions

Example: JSL Appliances
Using past data on TV sales, a tabular representation
of the probability distribution for sales was developed.
Units Sold
0
1
2
3
4
Number
of Days
80
50
40
10
20
200
x
0
1
2
3
4
f(x)
.40
.25
.20
.05
.10
1.00
80/200
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
11
Discrete Probability Distributions

Example: JSL Appliances
Graphical
representation
of probability
distribution
Probability
.50
.40
.30
.20
.10
0
1
2
3
4
Values of Random Variable x (TV sales)
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
12
Discrete Probability Distributions
In addition to tables and graphs, a formula that
gives the probability function, f(x), for every value
of x is often used to describe the probability
distributions.
Several discrete probability distributions specified
by formulas are the discrete-uniform, binomial,
Poisson, and hypergeometric distributions.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
13
Discrete Uniform Probability Distribution
The discrete uniform probability distribution is the
simplest example of a discrete probability
distribution given by a formula.
The discrete uniform probability function is
f(x) = 1/n
where:
the values of the
random variable
are equally likely
n = the number of values the random
variable may assume
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
14
Expected Value
The expected value, or mean, of a random variable
is a measure of its central location.
E(x) =  = xf(x)
The expected value is a weighted average of the
values the random variable may assume. The
weights are the probabilities.
The expected value does not have to be a value the
random variable can assume.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
15
Variance and Standard Deviation
The variance summarizes the variability in the
values of a random variable.
Var(x) =  2 = (x - )2f(x)
The variance is a weighted average of the squared
deviations of a random variable from its mean.
The weights are the probabilities.
The standard deviation, , is defined as the
positive square root of the variance.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
16
Expected Value

Example: JSL Appliances
x
0
1
2
3
4
f(x)
xf(x)
.40
.00
.25
.25
.20
.40
.05
.15
.10
.40
E(x) = 1.20
expected number of
TVs sold in a day
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
17
Variance

Example: JSL Appliances
x
x-
0
1
2
3
4
-1.2
-0.2
0.8
1.8
2.8
(x - )2
f(x)
(x - )2f(x)
1.44
0.04
0.64
3.24
7.84
.40
.25
.20
.05
.10
.576
.010
.128
.162
.784
TVs
squared
Variance of daily sales =  2 = 1.660
Standard deviation of daily sales = 1.2884 TVs
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
18
Using Excel to Compute the Expected
Value, Variance, and Standard Deviation

1
2
3
4
5
6
7
8
9
10
Excel Formula Worksheet
A
Sales
0
1
2
3
4
B
Probability
0.40
0.25
0.20
0.05
0.10
C
Sq.Dev.from Mean
=(A2-$B$8)^2
=(A3-$B$8)^2
=(A4-$B$8)^2
=(A5-$B$8)^2
=(A6-$B$8)^2
Mean =SUMPRODUCT(A2:A6,B2:B6)
Variance =SUMPRODUCT(C2:C6,B2:B6)
Std.Dev. =SQRT(B9)
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
19
Using Excel to Compute the Expected
Value, Variance, and Standard Deviation

1
2
3
4
5
6
7
8
9
10
Excel Value Worksheet
A
Sales
0
1
2
3
4
B
Probability
0.40
0.25
0.20
0.05
0.10
C
Sq.Dev.from Mean
1.44
0.04
0.64
3.24
7.84
Mean 1.2
Variance 1.66
Std.Dev. 1.2884
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
20
Binomial Probability Distribution

Four Properties of a Binomial Experiment
1. The experiment consists of a sequence of n
identical trials.
2. Two outcomes, success and failure, are possible
on each trial.
3. The probability of a success, denoted by p, does
not change from trial to trial.
stationarity
assumption
4. The trials are independent.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
21
Binomial Probability Distribution
Our interest is in the number of successes
occurring in the n trials.
We let x denote the number of successes
occurring in the n trials.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
22
Binomial Probability Distribution

Binomial Probability Function
𝑛!
𝑓 𝑥 =
𝑝 𝑥 (1 − 𝑝)(𝑛−𝑥)
𝑥! 𝑛 − 𝑥 !
where:
x = the number of successes
p = the probability of a success on one trial
n = the number of trials
f(x) = the probability of x successes in n trials
n! = n(n – 1)(n – 2) ….. (2)(1)
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
23
Binomial Probability Distribution

Binomial Probability Function
𝑛!
𝑓 𝑥 =
𝑝 𝑥 (1 − 𝑝)(𝑛−𝑥)
𝑥! 𝑛 − 𝑥 !
Number of experimental
outcomes providing exactly
x successes in n trials
Probability of a particular
sequence of trial outcomes
with x successes in n trials
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
24
Binomial Probability Distribution

Example: Evans Electronics
Evans Electronics is concerned about a low
retention rate for its employees. In recent years,
management has seen a turnover of 10% of the
hourly employees annually.
Thus, for any hourly employee chosen at random,
management estimates a probability of 0.1 that the
person will not be with the company next year.
Choosing 3 hourly employees at random, what is
the probability that 1 of them will leave the company
this year?
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
25
Binomial Probability Distribution

Example: Evans Electronics
The probability of the first employee leaving and the
second and third employees staying, denoted (S, F, F),
is given by
p(1 – p)(1 – p)
With a .10 probability of an employee leaving on any
one trial, the probability of an employee leaving on
the first trial and not on the second and third trials is
given by
(.10)(.90)(.90) = (.10)(.90)2 = .081
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
26
Binomial Probability Distribution

Example: Evans Electronics
Two other experimental outcomes result in one success
and two failures. The probabilities for all three
experimental outcomes involving one success follow.
Experimental
Outcome
Probability of
Experimental Outcome
(S, F, F)
(F, S, F)
(F, F, S)
p(1 – p)(1 – p) = (.1)(.9)(.9) = .081
(1 – p)p(1 – p) = (.9)(.1)(.9) = .081
(1 – p)(1 – p)p = (.9)(.9)(.1) = .081
Total = .243
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
27
Binomial Probability Distribution

Example: Evans Electronics
Let: p = .10, n = 3, x = 1
𝑛!
𝑓 𝑥 =
𝑝 𝑥 (1 − 𝑝)(𝑛−𝑥)
𝑥! 𝑛 − 𝑥 !
𝑓 1 =
3!
1! 3−1 !
0.1 1 (0.9)2 =
Using the
probability
function
.243
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
28
Binomial Probability Distribution

Example: Evans Electronics
1st Worker
2nd Worker
Using a tree diagram
3rd Worker
L (.1)
x
3
Prob.
.0010
S (.9)
2
.0090
L (.1)
2
.0090
S (.9)
1
.0810
L (.1)
2
.0090
S (.9)
1
.0810
1
.0810
0
.7290
Leaves (.1)
Leaves
(.1)
Stays (.9)
Leaves (.1)
Stays
(.9)
L (.1)
Stays (.9)
S (.9)
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
29
Using Excel to Compute
Binomial Probabilities

Excel Formula Worksheet
A
1
2
3
4
5
6
7
8
9
x
0
1
2
3
B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
f (x )
=BINOM.DIST(A5,$A$1,$A$2,FALSE)
=BINOM.DIST(A6,$A$1,$A$2,FALSE)
=BINOM.DIST(A7,$A$1,$A$2,FALSE)
=BINOM.DIST(A8,$A$1,$A$2,FALSE)
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
30
Using Excel to Compute
Binomial Probabilities

Excel Value Worksheet
A
1
2
3
4
5
6
7
8
9
x
0
1
2
3
B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
f (x )
0.729
0.243
0.027
0.001
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
31
Using Excel to Compute
Cumulative Binomial Probabilities

Excel Formula Worksheet
A
1
2
3
4
5
6
7
8
9
x
0
1
2
3
B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
Cumulative Probability
=BINOM.DIST(A5,$A$1,$A$2,TRUE )
=BINOM.DIST(A6,$A$1,$A$2,TRUE )
=BINOM.DIST(A7,$A$1,$A$2,TRUE )
=BINOM.DIST(A8,$A$1,$A$2,TRUE )
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
32
Using Excel to Compute
Cumulative Binomial Probabilities

Excel Value Worksheet
A
1
2
3
4
5
6
7
8
9
x
0
1
2
3
B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
Cumulative Probability
0.729
0.972
0.999
1.000
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
33
Binomial Probabilities
and Cumulative Probabilities
Statisticians have developed tables that give
probabilities and cumulative probabilities for a
binomial random variable.
These tables can be found in some statistics
textbooks.
With modern calculators and the capability of
statistical software packages, such tables are
almost unnecessary.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
34
Binomial Probability Distribution

Using Tables of Binomial Probabilities
p
n
x
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
3
0
1
2
3
.8574
.1354
.0071
.0001
.7290
.2430
.0270
.0010
.6141
.3251
.0574
.0034
.5120
.3840
.0960
.0080
.4219
.4219
.1406
.0156
.3430
.4410
.1890
.0270
.2746
.4436
.2389
.0429
.2160
.4320
.2880
.0640
.1664
.4084
.3341
.0911
.1250
.3750
.3750
.1250
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
35
Binomial Probability Distribution

Expected Value
E(x) =  = np

Variance
Var(x) = s 2 = np(1 – p)

Standard Deviation
𝜎=
𝑛𝑝(1 − 𝑝)
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
36
Binomial Probability Distribution

Example: Evans Electronics
• Expected Value
E(x) = np = 3(.1) = .3 employees out of 3
• Variance
Var(x) = np(1 – p) = 3(.1)(.9) = .27
•
Standard Deviation
𝜎=
3 .1 . 9) = .52 employees
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
37
Poisson Probability Distribution
A Poisson distributed random variable is often
useful in estimating the number of occurrences
over a specified interval of time or space
It is a discrete random variable that may assume
an infinite sequence of values (x = 0, 1, 2, . . . ).
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
38
Poisson Probability Distribution
Examples of Poisson distributed random variables:
the number of knotholes in 14 linear feet of
pine board
the number of vehicles arriving at a toll
booth in one hour
Bell Labs used the Poisson distribution to model
the arrival of phone calls.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
39
Poisson Probability Distribution

Two Properties of a Poisson Experiment
1. The probability of an occurrence is the same
for any two intervals of equal length.
2. The occurrence or nonoccurrence in any
interval is independent of the occurrence or
nonoccurrence in any other interval.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
40
Poisson Probability Distribution

Poisson Probability Function
𝜇 𝑥 𝑒 −𝜇
𝑓 𝑥 =
𝑥!
where:
x = the number of occurrences in an interval
f(x) = the probability of x occurrences in an interval
 = mean number of occurrences in an interval
e = 2.71828
x! = x(x – 1)(x – 2) . . . (2)(1)
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
41
Poisson Probability Distribution

Poisson Probability Function
Since there is no stated upper limit for the number
of occurrences, the probability function f(x) is
applicable for values x = 0, 1, 2, … without limit.
In practical applications, x will eventually become
large enough so that f(x) is approximately zero
and the probability of any larger values of x
becomes negligible.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
42
Poisson Probability Distribution

Example: Mercy Hospital
Patients arrive at the emergency room of Mercy
Hospital at the average rate of 6 per hour on
weekend evenings.
What is the probability of 4 arrivals in 30 minutes
on a weekend evening?
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
43
Poisson Probability Distribution

Example: Mercy Hospital
 = 6/hour = 3/half-hour, x = 4
𝑓 4 =
34 (2.71828)−3
4!
Using the
probability
function
= .1680
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
44
Using Excel to Compute
Poisson Probabilities

Excel Formula Worksheet
A
1
2
3
4
5
6
7
8
9
10
B
3 = Mean No. of Occurrences ()
Number of
Arrivals (x )
0
1
2
3
4
5
6
… and so on
Probability f (x )
=POISSON.DIST(A4,$A$1,FALSE)
=POISSON.DIST(A5,$A$1,FALSE)
=POISSON.DIST(A6,$A$1,FALSE)
=POISSON.DIST(A7,$A$1,FALSE)
=POISSON.DIST(A8,$A$1,FALSE)
=POISSON.DIST(A9,$A$1,FALSE)
=POISSON.DIST(A10,$A$1,FALSE)
… and so on
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
45
Using Excel to Compute
Poisson Probabilities

Excel Value Worksheet
A
1
2
3
4
5
6
7
8
9
10
B
3 = Mean No. of Occurrences ()
Number of
Arrivals (x )
0
1
2
3
4
5
6
… and so on
Probability f (x )
0.0498
0.1494
0.2240
0.2240
0.1680
0.1008
0.0504
… and so on
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
46
Poisson Probability Distribution

Example: Mercy Hospital
Poisson Probabilities
Probability
0.25
0.20
Actually,
the sequence
continues:
11, 12, 13 …
0.15
0.10
0.05
0.00
0
1
2
3
4
5
6
7
8
9
10
Number of Arrivals in 30 Minutes
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
47
Using Excel to Compute
Cumulative Poisson Probabilities

Excel Formula Worksheet
A
1
2
B
3 = Mean No. of Occurrences ( )
Number of
3 Arrivals (x )
4
0
5
1
6
2
7
3
8
4
9
5
10
6
… and so on
Cumulative Probability
=POISSON.DIST(A4,$A$1,TRUE)
=POISSON.DIST(A5,$A$1,TRUE)
=POISSON.DIST(A6,$A$1,TRUE)
=POISSON.DIST(A7,$A$1,TRUE)
=POISSON.DIST(A8,$A$1,TRUE)
=POISSON.DIST(A9,$A$1,TRUE)
=POISSON.DIST(A10,$A$1,TRUE)
… and so on
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
48
Using Excel to Compute
Cumulative Poisson Probabilities

Excel Value Worksheet
A
1
2
3
4
5
6
7
8
9
10
B
3 = Mean No. of Occurrences ( )
Number of
Arrivals (x)
0
1
2
3
4
5
6
… and so on
Cumulative Probability
0.0498
0.1991
0.4232
0.6472
0.8153
0.9161
0.9665
… and so on
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
49
Poisson Probability Distribution
A property of the Poisson distribution is that
the mean and variance are equal.
=2
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
50
Poisson Probability Distribution

Example: Mercy Hospital
Variance for Number of Arrivals
During 30-Minute Periods
=2=3
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
51
Hypergeometric Probability Distribution
The hypergeometric distribution is closely related
to the binomial distribution.
However, for the hypergeometric distribution:
the trials are not independent, and
the probability of success changes from trial
to trial.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
52
Hypergeometric Probability Distribution

Hypergeometric Probability Function
𝑓 𝑥 =
where:
𝑟
𝑥
𝑁−𝑟
𝑛−𝑥
𝑁
𝑛
x = number of successes
n = number of trials
f(x) = probability of x successes in n trials
N = number of elements in the population
r = number of elements in the population
labeled success
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
53
Hypergeometric Probability Distribution

Hypergeometric Probability Function
𝑓 𝑥 =
𝑟
𝑥
𝑁−𝑟
𝑛−𝑥
𝑁
𝑛
for 0 < x < r
number of ways
n – x failures can be selected
from a total of N – r failures
in the population
number of ways
x successes can be selected
from a total of r successes
in the population
number of ways
n elements can be selected
from a population of size N
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
54
Hypergeometric Probability Distribution

Hypergeometric Probability Function
The probability function f(x) on the previous slide
is usually applicable for values of x = 0, 1, 2, … n.
However, only values of x where: 1) x < r and
2) n – x < N – r are valid.
If these two conditions do not hold for a value of
x, the corresponding f(x) equals 0.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
55
Hypergeometric Probability Distribution

Example: Neveready’s Batteries
Bob Neveready has removed two dead batteries
from a flashlight and inadvertently mingled them
with the two good batteries he intended as
replacements. The four batteries look identical.
Bob now randomly selects two of the four
batteries. What is the probability he selects the two
good batteries?
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
56
Hypergeometric Probability Distribution

Using the
probability
function
Example: Neveready’s Batteries
𝑓 𝑥 =
𝑟
𝑥
𝑁−𝑟
𝑛−𝑥
𝑁
𝑛
=
2 2
2 0
4
2
=
2!
2!0!
2!
0!2!
4!
2!2!
1
6
= = .167
where:
x = 2 = number of good batteries selected
n = 2 = number of batteries selected
N = 4 = number of batteries in total
r = 2 = number of good batteries in total
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
57
Using Excel to Compute
Hypergeometric Probabilities

Excel Formula Worksheet
A
1
2
3
4
5
6
7
B
2
2
2
4
Number of Successes (x )
Number of Trials ( n )
Number of Elements in the Population Labeled Success ( r )
Number of Elements in the Population (N )
f (x ) =HYPGEOM.DIST(A1,A2,A3,A4)
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
58
Using Excel to Compute
Hypergeometric Probabilities

Excel Value Worksheet
A
1
2
3
4
5
6
7
B
2
2
2
4
Number of Successes (x )
Number of Trials ( n )
Number of Elements in the Population Labeled Success ( r )
Number of Elements in the Population (N )
f (x ) 0.1667
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
59
Hypergeometric Probability Distribution

Mean
𝑟
𝐸 𝑥 =𝜇=𝑛
𝑁

Variance
𝑟
𝑉𝑎𝑟 𝑥 = 𝜎 = 𝑛
𝑁
2
𝑟
1−
𝑁
𝑁−𝑛
𝑁−1
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
60
Hypergeometric Probability Distribution

Example: Neveready’s Batteries
• Mean
𝑟
2
𝜇=𝑛
=2
= 1
𝑁
4
•
Variance
𝜎2
2
=2
4
2
1−
4
4−2
1
= = .333
4−1
3
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
61
Hypergeometric Probability Distribution
Consider a hypergeometric distribution with n trials
and let p = (r/n) denote the probability of a success
on the first trial.
If the population size is large, the term
(N – n)/(N – 1) approaches 1.
The expected value and variance can be written
E(x) = np and Var(x) = np(1 – p).
Note that these are the expressions for the expected
value and variance of a binomial distribution.
continued
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
62
Hypergeometric Probability Distribution
When the population size is large, a hypergeometric
distribution can be approximated by a binomial
distribution with n trials and a probability of
success p = (r/N).
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
63
End of Chapter 5
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
64