Business Analytics: Methods, Models,

and Decisions, 1st edition

James R. Evans

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Copyright © 2013 Pearson Education, Inc.
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 Basic Concepts of Probability
 Random Variables and Probability Distributions
 Discrete Probability Distributions
 Continuous Probability Distributions
 Random Sampling from Probability Distributions
 Data Modeling and Distribution Fitting

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 Probability is the likelihood that an outcome
occurs.
 An experiment is the process that results in an
outcome.
 The outcome of an experiment is a result that we
observe.
 The sample space is the collection of all possible
outcomes of an experiment.
 An event is a collection of one or more outcomes
from a sample space.

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Probabilities may be defined from one of three
perspectives:
 Classical definition: probabilities can be deduced
from theoretical arguments
 Relative frequency definition: probabilities are
based on empirical data
 Subjective definition: probabilities are based on
judgment and experience

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Example 5.1 Classical Definition of Probability
Suppose we roll 2 dice
 Probability die rolls sum to three = 2/36

Suppose two consumers try a new product.

 Assume equally likely possible outcomes:
1. like, like
2. like, dislike
3. dislike, like
4. dislike, dislike
 Probability at least one dislikes product = 3/4

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Example 5.2
Relative Frequency Definition of Probability
 Probability a computer is repaired in 10 days
= 0.076

Figure 5.1

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Two Basic Facts Govern Probability
 The probability associated with any outcome must
be between 0 and 1.
0 ≤ P(Oi)≤1 for each outcome Oi

The sum of the probabilities over all possible

outcomes must be equal to 1.
P(O1) + P(O2) + … + P(On) = 1
(n is the number of outcomes in the sample space.)

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Example 5.3
Computing the Probability of an Event
Consider the events:
 Rolling 7 or 11 on two dice
Probability = 6/36 + 2/36 = 8/36.
 Repair a computer in 7 days or less
Probability =
= O1 + O2 +O3 +O4 +O5 +O6 +O7
= 0 + 0 + 0 + 0 + .004 + .008 + .002
= 0.032
From Figure 5.1

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Example 5.4 Computing the Probability of the
Complement of an Event
 Ac, the complement of A, consists of all outcomes
in the sample space not in A.
Dice example:
 A = {7, 11}
P(A) = 8/36
 Ac = {2, 3, 4, 5, 6, 8, 9, 10, 12}
P(Ac) = 1 − 8/36 = 28/36

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Example 5.5 Computing the Probability of Mutually
Exclusive Events
 Mutually exclusive events have no outcomes in
common.
Dice Example:
 A = {7, 11}
 B = {2, 3, 12}
 P(A or B) = UNION of events A and B
= P(A) + P(B)
= 8/36 + 4/36 = 12/36
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Example 5.6 Computing the Probability of Non-
Mutually Exclusive Events

Dice Example:
 A = {2, 3, 12}
 B = {even number}
 P(A or B) = UNION of events A and B
= P(A) + P(B) − P(A andB)
= 4/36 + 18/36 − 2/36
= 20/36
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Example 5.7 Conditional Probability in Marketing
 The Data shows the first and
second purchases for a
sample of 200 customers.
 Probability of purchasing an
iPad given already purchased Figure 5.2

an iMac = 2/13

Figure 5.3

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Example 5.8 Computing a Conditional Probability in
a Cross-Tabulation
 Probability of preferring Brand 1 given that a
respondent is male = 25/63

Figure 5.4

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Example 5.9
Using the Conditional Probability Formula

 Probability of A given B:

 P(B1|M) = P(B1 and M)/P(M)

= (25/100)/(63/100)
= 25/63 = 0.397
Summary of conditional probabilities:

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Example 5.10
Using the Multiplication Law of Probability

Texas Hold ‘Em Poker Game

 Probability of pocket aces (two aces in hand):
 P(Ace on first card and Ace on second card)
= P(A1 and A2)
= P(A2|A1) P(A1)
= (3/51) (4/52)
= 0.004525

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Example 5.11 Determining if Two Events are
Independent
 Are Gender and Brand Preference Independent?

 Is P(B1|M) =P(B1)?
0.397 ≠ .34
Gender and Brand Preference are Dependent.
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Example 5.12
Using the Multiplication Law for Independent Events

 Dice Roll Example:

 Rolling pairs of dice are independent events since
they do not depend on the previous rolls.
 A = {roll a sum of 6 on first pair die rolls}
 B = {roll a sum of 2, 3, or 12 on second pair rolls}
 P(A and B) = P(A) P(B)
= (5/36) (4/36) = 0.0154
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 A random variable is a numerical description of the
outcome of an experiment.
 A discrete random variable is one for which the
number of possible outcomes can be counted.
 A continuous random variable has outcomes over
one or more continuous intervals of real numbers.
 A probability distribution is a characterization of
the possible values that a random variable may
assume along with the probability of assuming
these values.

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Example 5.13
Discrete and Continuous Random Variables
Examples of Discrete Variables:
 outcomes of dice rolls
 whether a customer likes or dislikes a product
 number of hits on a Web site link today
Examples of Continuous Variables:
 weekly change in DJIA
 daily temperature
 time between machine failures

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Example 5.14
Probability Distribution of Dice Rolls

Figure 5.5

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Empirical Probability Distribution - Sample data is
used to approximate a probability distribution

Figure 5.1

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Example 5.15 A Subjective Probability Distribution
 Distribution of an expert’s assessment of how the
DJIA might change next year.

Figure 5.6

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Probability Mass Function
a mathematical function f(x) specifying the
probability of the random variable X.
xi represents the i th value of X.
Properties:

Cumulative distribution function:

F(x) = P(X ≤ x)

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Example 5.16
Probability Mass Function for Rolling Two Dice
f(x2) = 1/36
f(x3) = 2/36
f(x4) = 3/36
f(x5) = 4/36
f(x6) = 5/36
:
f(x12) = 1/36
Figure 5.5

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Example 5.17
Using the Cumulative Distribution Function
 Probability of rolling between 4 and 8:
= P(4 ≤ X ≤ 8)
= P(3 <X ≤ 8)
= F(x8) – F(x3)
=13/18 – 1/12
= 23/36

Figure 5.7

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Example 5.18 Computing the Expected Value
of the sum of values on 2 die rolls

E[X] = 2(1/36) + 3(1/18) + …

12(1/36) = 7

Figure 5.8

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Example 5.19 Expected Value on Television
Apprentice example
 Teams were required to select an artist
(mainstream or avant-garde) and sell their art for
the most money possible.
Deal or No Deal example
 Contestant had 5 briefcases left with $100, $400,
$1000, $50,000 or $300,000 in them.
 Expected value of briefcases is $70,300.
 Banker offered contestant $80,000 to quit.

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Example 5.20 Expected Value of Charitable Raffle
 Cost of raffle ticket is $50
 1000 raffle tickets were sold.
 Prize for winning raffle is $25,000

E[X] = −$25

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Example 5.21 Airline Revenue Management
 Full and discount airfares are available for a flight.
 Full-fare ticket costs $560

 Discount ticket costs $400

 X = ticket price paid
 p = 0.75 (the probability of selling a full-fare ticket)
 E[X] = 0.75($560) + 0.25(0) = $420

 The airline should not discount full-fare tickets

because the expected value of a full-fare ticket is
greater than the cost of a discount ticket.
 Break-even point: $400 = p($560) or p = 0.714

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Example 5.22
Computing the Variance of a Random Variable

Figure 5.9

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Bernoulli Distribution
 two possible outcomes each with a constant
probability of occurrence
 typically “success” is x = 1 and “failure” x = 0
 p is the probability of a success outcome

E[X] = p
Var[X] = p(1 −p)

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Example 5.23 Using the Bernoulli Distribution
Model whether an individual responds positively to a
telemarketing promotion.
 You have a box with 20 red and 80 white marbles.
 You ask individuals exposed to the telemarketing
promotion to select a marble and then replace it.
 If the customer selects a red marble, the customer
makes a purchase.
 If the customer selects a white marble, the
customer does not make a purchase.

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Binomial Distribution
 Models n independent replications of a Bernoulli
experiment
 X represents the number of successes in these n
experiments

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Example 5.24 Computing Binomial Probabilities
 Suppose 10 individuals receive the telemarking
promotion.
 Each individual has a 0.2 probability of making a
purchase.
 Find the probability that exactly 3 of the 10
individuals make a purchase.

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Example 5.25
True: F(x)
Using Excel’s Binomial Distribution Function
False: f(x)

P(x = 3) = 0.20133
= f(3)
=BINOM.DIST(3, 10, 0.2, true)
P(x≤ 3) = 0.87913
= F(3)
=BINOM.DIST(3, 10, 0.2, false)

Figure 5.10

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Histogram Example of the Binomial Distribution
Symmetric when p = 0.5

Figure 5.11a

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Histogram Examples of the Binomial Distribution

Positively skewed when p< 0.5 Negatively skewed when p< 0.5

Figure 5.11b Figure 5.11c

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Poisson Distribution
 Models the number of occurrences in some unit of
measure (often time or distance).
 There is no limit on the number of occurrences.
 The average number of occurrence per unit is a
constant denoted as λ.

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Example 5.26 Computing Poisson Probabilities
 Suppose the average number of customers
arriving at a Subway restaurant during lunch hour
is λ =12 per hour.
 The probability that exactly x customers arrive
during the hour is given by the Poisson
distribution.
 Find the probability that exactly 5 arrive during
lunch hour:
 f(5) = e-12(125)/5!
= (0.000006144)(248,832)/120
= 0.1274
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Example 5.27 Using Excel’s True: F(x)
Poisson Distribution Function False: f(x)

POISSON.DIST(x, mean, cumulative)

Figure 5.13
Figure 5.12

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Analytics in Practice: Using the Poisson Distribution
for Modeling Bids on Priceline
 Pricing strategies for Kimpton
hotels on Priceline is modeled
using a Poisson distribution.
 The number of bids placed
per day 3 days before arrival
is f(x) = e-6.3(6.3x)/x! .
 Using the model increased
sales 11% in one year.

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 Change in
DJIA using
5% increments
Figure 5.6

Change in
DJIA using
2.5%
increments

Approaching a
Figure 5.14 smooth curve
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Probability density function
 A curve described by a mathematical function that
characterizes a continuous random variable
Properties of a probability density function
 f(x)≥ 0 for all values of x
 Total area under the density function equals 1.
 P(X = x) = 0
 Probabilities are only defined over an interval.
 P(a ≤ X≤ b) is the area under the density function
between a and b.

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Uniform Distribution
 All outcomes between a minimum (a) and a
maximum (b) are equally likely.

Area = 1

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Uniform Distribution

 Expected Value =

 Variance =

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Example 5.28
Computing Uniform Probabilities
 Sales revenue for a product varies uniformly each
week between $1000 and $2000.
 f(x) = 1/(2000-1000)
= 1/1000

Area = 1

Figure 5.15

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Example 5.28 (continued)
Computing Uniform Probabilities
 Find the probability sales revenue will be less than
$1,300.
 P(X< 1300) = (1300-1000)(1/1000) = 0.30

Figure 5.16

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Example 5.28 (continued) Uniform Probabilities
 Find the probability that revenue will be between
$1,500 and $1,700.

Figure 5.17

 P(1500 ≤X≤ 1700) = P(X≤1700) − P(X≤1500)

= F(1700) − F(1500)
= 300/1000 − 500/1000
=0.20
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Normal Distribution
- f(x) is a bell-shaped curve
- Characterized by 2 parameters
 (mean)
 (standard deviation)
- Properties
1. Symmetric
2. Mean = Median = Mode
3. Unbounded
4. Empirical rules apply

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 Figure 5.18

True: F(x)
=NORM.DIST(x, mean, stdev, cumulative) False: f(x)

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Example 5.29 Using NORM.DIST to Compute
Normal Probabilities
 The distribution for customer demand (units per
month) is normal with:
mean = 750
stdev. = 100
 Find the probability that demand will be:
a) at most 900 units/month
b) exceed 700 units/month
c) be between 700 and 900 units/month

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Example 5.29 (continued) Using
NORM.DIST to Compute
Normal Probabilities
Cumulative probabilities
are computed as:
=NORM.DIST(x, 750, 100, true)

a) P(X < 900) = 0.9332

b) P(X> 700) = 1−0.3085 = 0.6915
Figure 5.19
c) P(700 <X< 900) = 0.9332−0.3085
= 0.6247

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Example 5.29 (continued) Using NORM.DIST to
Compute Normal Probabilities

0.9332

Figure 5.20a

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Example 5.29 (continued) Using NORM.DIST to
Compute Normal Probabilities

0.6915

Figure 5.20b

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Example 5.29 (continued) Using NORM.DIST to
Compute Normal Probabilities

0.6247

Figure 5.20c

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Example 5.30 Using the NORM.INV Function
=NORM.INV(probability, mean, stdev)
provides the x value with F(x)= probability

 What level of demand would be exceeded at most

10% of the time?
 Find x such that F(x) = 90%
= NORM.INV(0.90, 750, 100)
results in x = 878.155

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Example 5.30 (continued)
Using the NORM.INV Function

90%

878.155 Figure 5.20d

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Standard Normal Distribution
Z is the standard normal random variable with:
Mean = 0
Stdev = 1

Figure 5.21

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Example 5.31 Computing Probabilities with the
Standard Normal Distribution
 Verify the empirical rules using Excel.
 P(-1 <Z< 1 )
= NORMS.DIST(1) – NORMS.DIST(-1)
= 0.84134 – 0.15866
= 0.6827
~ 68%

Figure 5.22
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Example 5.32 Computing Probabilities with the
Standard Normal Tables
 From Example 5.29, what is the probability that
demand will be at least 900 units/month?
 Use the equation:
 Z = (900 − 750)/100
= 1.50
 Using Table 1 in Appendix B, we find:
P(X< 900) = P(Z< 1.50)
= 0.93319

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Exponential Distribution
 Models the time between randomly occurring
events (arrivals, machine failures, etc.)

with λ=1

 where λ is the mean

rate of occurrences
(from the discrete
Poisson distribution) Figure 5.23

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Example 5.33 Using the Exponential Distribution
 The mean time to failure of a critical engine
component is µ = 8,000 hours.
What is the probability of failing before 5000 hours?
 P(X < x) =EXPON.DIST(x, lambda, cumulative)
 Since , we can solve for
λ = 1/8000
P(x< 5000) =EXPON.DIST(5000, 1/8000, true)
= 0.4647

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Example 5.33 (continued)
Using the Exponential Distribution

Figure 5.24

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Other Useful Continuous
Distributions
 Triangular Distribution
 Lognormal Distribution
 Beta Distribution

Figure 5.25

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 Many application in Business Analytics require
random samples from specific probability
distributions.
 For example, cumulative discounted cash flow can
be described using probability distributions for
sales, sales growth rate, operating expenses, and
inflation factors (all uncertain variables).
 Random numbers are the basis for generating
random samples from probability distributions.
 Random numbers are uniformly distributed on the
interval from 0 to 1.

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Example 5.34 Sampling from the Distribution of
Dice Outcomes
Probability distribution Intervals for random sampling

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Example 5.34 (continued) Sampling from the
Distribution of Dice Outcomes
=RAND( ) generates random numbers in Excel

Outcome = 8 since 0.681 is

between 0.583 and 0.722

Outcome = 4 since 0.119 is

between 0.083 and 0.167

Figure 5.26

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Example 5.35 Using the VLOOKUP Function
 Generate a random sample of Changes in DJIA.
 First compute F(x)
 Assign intervals to outcomes
 Generate random numbers using =RAND( )
=VLOOKUP(H2, $E2:$G$10, 3)

Figure 5.27

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Example 5.36
Using Excel’s Random Number Generation Tool
Generate 100 outcomes from a Poisson distribution with a
mean of 12.
Data
Data Analysis
Random Number Generation
Number of Variables: 1
Number of Random Numbers: 100
Distribution: Poisson
Parameter: Lambda = 12

Figure 5.28

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Example 5.36 (continued) Using Excel’s Random
Number Generation Tool
Histogram of 100 random outcomes

Figure 5.29

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Probability Distributions available in Excel’s Data
Analysis Toolpak under Random Number Generation
 Bernoulli
 Binomial
 Discrete
 Normal
 Patterned
 Poisson
 Uniform

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Sampling From Various Probability Distributions
Using Standard Excel Functions
Uniform: =RANDBETWEEN(a, b)
You can also replace probability with RAND( ) in
many inverse probability distribution functions:
Binomial: =BINOM.INV(trials, p, RAND( ))
Normal: =NORM.INV(RAND( ), mean, stdev)
Logormal: =LOGNORM.INV(RAND( ), mean, stdev)
Beta: =BETA.INV(RAND( ), alpha, beta, A, B)

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Example 5.37 A Sampling Experiment for
Evaluating Capital Budgeting Projects
 In finance, one way of evaluating capital budgeting
projects is to compute:
 Profitability indexPI = PV / I
where PV is the present value of future cash flows
and I is the initial investment

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Example 5.37 (continued) A Sampling Experiment
for Evaluating Capital Budgeting Projects
 Suppose PV is normally distributed with a mean of
$12 million and standard deviation of $2.5 million.
 I is also normally distributed with a mean of $3
million and standard deviation of $0.8 million.
 Use a sampling experiment to generate a
probability distribution for PI = PV / I.

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Example 5.37 (continued)
 For PV: =NORM.INV(RAND( ), 12, 2.5)
 For I: =NORM.INV(RAND( ), 3, 0.8)

PI = PV/I

Profitability
Index mean
= 4.76

Figure 5.30

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Example 5.37 (continued) A Sampling Experiment
for Evaluating Capital Budgeting Projects
 Profitability Index is skewed to the right

Figure 5.31

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Risk Solver Platform
Probability Distribution Functions

Table 5.1

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Example 5.38 Using Risk Solver Platform
Distribution Functions
 An energy company is considering offering a new
product and needs to estimate the growth in PC
ownership. The expected growth rates are:
 Minimum = 5%
 Most likely = 7.7%
 Maximum =10%

 Generate 500 samples of PC ownership growth

rate using: =PsiTriangular(5%, 7.7%, 10%)

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Example 5.38 (continued) Using Risk Solver
Platform Distribution Functions

Figure 5.32

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Example 5.39 Analyzing Airline Passenger Data
 Sample data on passenger demand for 25 flights

Can we assume normally distributed?

Figure 5.33

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Example 5.40 Analyzing Airport Service Times
 Sample data on service times for 812 passengers
at an airport’s ticketing counter

Can we assume normally distributed? Figure 5.34

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Goodness of Fit
 The basis for fitting data to a probability
distribution
 Attempts to draw conclusions about the nature of
the distribution
 Three statistics measure goodness of fit:
Chi-square
Kolmogorov-Smirnov
Anderson-Darling

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Example 5.41
Fitting a Distribution to Airport Service Times
1. Highlight the data
Risk Solver
Tools
Fit
2. Fit Options dialog
Type: Continuous
Test: Kolmorgov-Smirnov
Figure 5.35

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Example 5.41 (continued)
Fitting a Distribution to Airport Service Times
 Erlang is the best-fitting distribution

Figure 5.36
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Analytics in Practice: The Value of
Good Data Modeling in Advertising
 Gross’s model:
 A mathematical model that relates the relative
contributions of creative and media dollars to total
advertising effectiveness.
 Often used to identify the best number of ads to
purchase.
 Analysis found that the optimal number of ads can
vary significantly depending on the shape of the
distribution of effectiveness for a single ad.

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 Bernoulli distribution  Discrete uniform
 Binomial distribution distribution
 Complement  Empirical probability
 Conditional probability distribution
 Continuous random  Event
variable  Expected value
 Cumulative  Experiment
distribution function  Exponential distribution
 Discrete random  Goodness of fit
variable  Independent events

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 Multiplication law of  Probability mass
probability function
 Mutually exclusive  Random number
 Normal distribution  Random number seed
 Outcome  Random variable
 Poisson distribution  Random variate
 Probability  Sample space
 Probability density  Standard normal
function distribution
 Probability distribution  Uniform distribution
 Union
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 Recall that PLE produces lawnmowers and a
medium size diesel power lawn tractor.
 Determine probability distributions appropriate for
modeling mower failures and blade weights.
 Determine the sampling distribution of the mean
mower failures for a sample size of 100.
 Determine the sampling distribution of the
proportion blade weights below specification for a
sample size of 350.
 Analyze your results and write up a formal report.
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publishing as Prentice Hall 5-89
Copyright © 2013 Pearson Education, Inc.
publishing as Prentice Hall 5-90