118
CHAPTER 9
Theoretical Probability Models
Notes
Chapter 9 is a straightforward treatment of probability modeling using five standard distributions. The five
distributions (binomial, Poisson, exponential, normal, beta) were chosen because they cover a variety of
different probabilitymodeling situations. Tables of probabilities for four of the distributions (all except the
exponential, probabilities for which are easily calculated) are provided in the appendices.
In addition to the five distributions treated in the main part of the chapter, the uniform distribution is
developed in problems 9.27  9.29, the triangular distribution in 9.30, and the lognormal distribution in
problem 9.36 and the Municipal Solid Waste case study. Depending on the nature of the course and the
level of the students, instructors may wish to introduce other distributions.
One note of caution: Chapter 9 provides an introduction to probability distributions that are used in Chapter
10 (fitting model parameters and natural conjugate priors) as well as Chapter 11 (creating random variates
in a simulation). In particular, if the course is intended to move on to Chapter 11, it is important to expose
students to the uniform distribution, the triangular distribution, and some of the other distributions as well.
In addition to using the tables provided in the appendices, RISKview can be used for computing and seeing
distributions.
Stepbystep instructions for viewing theoretical distributions with RISKview are provided in
the chapter.
RISKview is a popup dialog box within @RISK that is activated using the Define
Distribution button.
The desired probabilities can be determined by sliding the delimiters (marked by
inverted triangles) or by entering Left and Right X and P values in the statistics grid.
Topical crossreference for problems
Bayes’ theorem
9.17, 9.21
Beta distribution
9.4, 9.5, 9.22, 9.24
Binomial distribution
9.1, 9.5, 9.15  9.19, 9.29, 9.31,
9.34, 9.35, Overbooking
Central limit theorem
9.36, Municipal Solid Waste
Empirical rule
9.11
Exponential distribution
9.5, 9.8, 9.12, 9.14, 9.15,
Earthquake Prediction
Linear transformations
9.14, 9.25, 9.26
Lognormal distribution
9.36, Municipal Solid Waste
Memoryless property
9.12, 9.13, 9.28
Normal distribution
9.2, 9.5  9.7, 9.11, 9.25, 9.26,
9.31, 9.36, Municipal Solid Waste
Pascal sampling
9.18
Poisson distribution
9.3, 9.5, 9.9, 9.10, 9.13, 9.15, 9.19
 9.21, 9.32  9.35, Earthquake
Prediction
PrecisionTree
9.35
Requisite models
9.23
RISKview
9.19.15, 9.18, 9.19, 9.22, 9.249.34
Sensitivity analysis
9.17, 9.24, Earthquake Prediction
Triangular distribution
9.30
Uniform distribution
9.27  9.29
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Solutions
9.1.
Find P(Gone 6 or more weekends out of 12)
=
P
B
(
R
≥
6 
n
= 12,
p
= 0.65)
=
P
B
(
R
'
≤
6 
n
= 12,
p
= 0.35) = 0.915.
Being gone 6 or more weekends out of 12 is the same as staying home on 6 or fewer.
Using RISKview, start RISKview.
Select a function as the distribution source, the binomial as the
distribution type, 12 for the n parameter, and 0.65 for the p parameter.
In the statistics grid, set the Left X
value to 5.5, and the Right X value to 12.
The Difference P value then shows the desired probability:
91.54%.
This distribution is saved as a RISKview file titled “Problem 9.1.rvp”.
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 Spring '11
 John

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