1. Slide Sets to © 2005 by McGraw-Hill,19-1
Developed By:
Dr. Don Smith, P.E.
Department of Industrial
Engineering
Texas A&M University
College Station, Texas
Executive Summary Version
Chapter 19
More on Variation and
Decision Making
Under Risk

2. Slide Sets to © 2005 by McGraw-Hill,19-2
LEARNING OBJECTIVESLEARNING OBJECTIVES
1. Certainty and
risk
2. Variables and
distributions
3. Random
samples
4. Average and
dispersion
5. Monte Carlo
simulation

3. Slide Sets to © 2005 by McGraw-Hill,19-3
Sct 19.1 Interpretation of Certainty, Risk,Sct 19.1 Interpretation of Certainty, Risk,
and Uncertaintyand Uncertainty
 Certainty – Everything know for sure; not present in
the real world of estimation, but can be ‘assumed’
 Risk – a decision making situation where all of the
outcomes are know and the associated probabilities
are defined
 Uncertainty – One has two or more observable values
but the probabilities associated with the values are
unknown
 Observable values – states of nature
 See Example 19.1 – about risk

4. Slide Sets to © 2005 by McGraw-Hill,19-4
Types of Decision MakingTypes of Decision Making
 Decision Making under Certainty
 Process of making a decision where all of the input
parameters are known or assumed to be known
 Outcomes – known
 Termed a deterministic analysis
 Parameters are estimated with certainty
 Decision Making under Risk
 Inputs are viewed as uncertain, and element of chance is
considered
 Variation is present and must be accounted for
 Probabilities are assigned or estimated
 Involves the notion of random variables

5. Slide Sets to © 2005 by McGraw-Hill,19-5
Two Ways to Consider Risk inTwo Ways to Consider Risk in
Decision MakingDecision Making
Expected Value (EV) analysis
Applies the notion of expected value (Chapter 18)
Calculation of EV of a given outcome
Selection of the outcome with the most
advantageous outcome
Simulation Analysis
Form of generating artificial data from assumed
probability distributions
Relies on the use of random variables and the laws
associated with the algebra of random variables

6. Slide Sets to © 2005 by McGraw-Hill,19-6
Sct 19.2 Elements Important to DecisionSct 19.2 Elements Important to Decision
Making Under RiskMaking Under Risk
 The concept of a random variable
A decision rule that assigns an outcome to a
sample space
Discrete variable or Continuous variable
 Discrete variable – finite number of outcomes possible
 Continuous variable – infinite number of outcomes
 Probability
Number between 0 and 1
Expresses the “chance” in decimal form that a
random variable will take on any specific value

7. Slide Sets to © 2005 by McGraw-Hill,19-7
Types of Random VariablesTypes of Random Variables
 Continuous
 Discrete

8. Slide Sets to © 2005 by McGraw-Hill,19-8
Distributions - Continuous VariablesDistributions - Continuous Variables
 Probability Distribution (pdf)
 A function that describes how
probability is distributed over the
different values of a variable
P(Xi) = probability that X = Xi
 Cumulative Distribution (cdf)
 Accumulation of probability over
all values of a variable up to and
including a specified value
F(Xi) = sum of all probabilities
through the value Xi
= P(X ≤ Xi)

9. Slide Sets to © 2005 by McGraw-Hill,19-9
Three Common Random VariablesThree Common Random Variables
 Uniform – equally likely
outcomes
 Triangular
 Normal
Study
Example 19.3

10. Slide Sets to © 2005 by McGraw-Hill,19-10
Discrete Density and Cumulative ExampleDiscrete Density and Cumulative Example
pdf cdf

11. Slide Sets to © 2005 by McGraw-Hill,19-11
Sct 19.3 Random SamplesSct 19.3 Random Samples
 Random Sample
A random sample of size n is the selection in a
random fashion with an assumed or known
probability distribution such that the values of the
variable have the same chance of occurring in the
sample as they appear in the population
Basis for Monte Carlo Simulation
 Can sample from:
Discrete distributions … or
Continuous distributions

12. Slide Sets to © 2005 by McGraw-Hill,19-12
Sampling from a Continuous DistributionSampling from a Continuous Distribution
 Form the cumulative
distribution in closed form
from the pdf
 Generate a uniform random
number on the interval
{0 – 1}, called U(0,1)
 Locate U(0,1) point on y-axis
 Map across to intersect the cdf
function
 Map down to read the outcome
(variable value) on x-axis

13. Slide Sets to © 2005 by McGraw-Hill,19-13
Sct 19.4 Expected Value and StandardSct 19.4 Expected Value and Standard
DeviationDeviation
 Two important parameters of a given random
variable:
Mean - µ
 Measure of central tendency
Standard Deviation - σ
 Measure of variability or spread
 Two Concepts to work within
Population
Sample from a population

14. Slide Sets to © 2005 by McGraw-Hill,19-14
Sample
Population vs SamplePopulation vs Sample
Population
 µ - population mean
 σ2
- population
variance
 σ - population
standard deviation
 Often sample from a
population in order to
make estimates
X
2
s
s
Sample mean
Sample variance
Sample standard
deviation
These values, properly sampled,
attempt to estimate their population
counterparts

15. Slide Sets to © 2005 by McGraw-Hill,19-15
Important RelationshipsImportant Relationships
Population Mean µ
 Distribution
 E(x) =
 Sample
 Measure of the central tendency of the population
 If one samples from a population the hope is that
sample mean is an unbiased estimator of the true,
but unknown, population mean
( )i iX P X∑
i i iX f X
n n
=
∑ ∑

16. Slide Sets to © 2005 by McGraw-Hill,19-16
Variance and Standard DeviationVariance and Standard Deviation
Notes relating to
variance and standard
deviation properties
Illustration of variances
for discrete and
continuous distributions

17. Slide Sets to © 2005 by McGraw-Hill,19-17
Population vs. SamplePopulation vs. Sample
 Variance of a population
 Standard deviation of a
population:
N
2
i
i=1
(x )
N
σ
µ−
=
∑
2
Var(X)X Xσ σ= =
N
2
i
2 i=1
(x )
N
σ
µ−
=
∑
 Variance of a sample
 Standard deviation of a sample
2
2 2
1 1
iX n
s X
n n
= −
− −
∑
S is termed an unbiased estimator of the
population standard deviation

18. Slide Sets to © 2005 by McGraw-Hill,19-18
Combining the Average and StandardCombining the Average and Standard
DeviationDeviation
 Determine the percentage or fraction of the sample
that is within ±1, ±2, ±3 standard deviations of the
sample mean . . .
 In terms of probability…
 Virtually all of the sample values will fall within the
±3s range of the sample mean
 See example 19.6
, t = 1,2,3X ts±
P( )X ts X X ts− ≤ ≤ +

19. Slide Sets to © 2005 by McGraw-Hill,19-19
Continuous Random VariablesContinuous Random Variables
 Expected value:
Variance:
 For uniform pdf in Example 19.3:
( )( )
R
Xf x dxE X = ∫
2 2
( ) ( ) [ ( )]
R
Var x X f X dX E X= −∫
R represents the defined range of the
variable in question.
10
2 15
R
2 2 3 15 2
10
1
0.2 $10 X $15
5
E(X)= (0.2)Xdx=0.1X 0.1(225 100) $12.50
0.2
( ) (0.2) (12.5) (12.5)
3
=0.06667(3375-1000)-156.25=2.08
2.08 $1.44
( )
R
X
Var X X dX X
f x
σ
= ≤ ≤
= − =
= − = −
= =
=
∫
∫

20. Slide Sets to © 2005 by McGraw-Hill,19-20
Sct 19.5 Monte Carlo Sampling andSct 19.5 Monte Carlo Sampling and
Simulation AnalysisSimulation Analysis
 Simulation involves the generation of artificial
data from a modeled system
 Monte Carlo Sampling
The generation of samples of size n for selected
parameters of formulated alternatives
The sampled parameters are expected to vary
according to a stated probability distribution
(assumed)

21. Slide Sets to © 2005 by McGraw-Hill,19-21
Key Assumption - IndependenceKey Assumption - Independence
 For a given problem:
All parameters are assumed to be independent
One variable’s distribution in no way impacts any
other variable’s distribution
Termed:
Property of independent random variables
 The modeling approach follows a 7-step
process (page 678)

22. Slide Sets to © 2005 by McGraw-Hill,19-22
SimulationSimulation
Monte Carlo Sampling is a traditional
approach (method) for generating pseudo-
random numbers (RN) to sample from a
prescribed probability distribution.
 Pseudo-random refers to the fact that a digital
computer can generate approximately random
numbers due to fixed word size and round off
problems.

23. Slide Sets to © 2005 by McGraw-Hill,19-23
Sampling ProcessSampling Process
 Requires:
 The cdf of the assumed pdf;
 A uniform random number generator;
 Application of the inverse transform approach.
 Why require the cdf?
 The y-axis of a cdf is scaled from 0 to 1.
 That is the same as the range of U(0,1).
 Facilitates mapping a RN to achieve the outcome value on
the x-axis.
 The U(0,1) selects a X-value from the cdf.

24. Slide Sets to © 2005 by McGraw-Hill,19-24
The Need for the cdfThe Need for the cdf
x-axis: Outcome values
Cumulative probability on the y-axis
Generate a
U(0,1) random
number: Locate
that value on the
y-axis: Map
across to the cdf
then map down
to the x-axis to
obtain the
outcome

25. Slide Sets to © 2005 by McGraw-Hill,19-25
Summary of the Modeling StepsSummary of the Modeling Steps
 Formulate the economic analysis:
The alternatives – if more than one;
Define which parameters are “constants” and
which are to be viewed as random variables.
For the random variables, assign the appropriate
pdf:
Discrete and or continuous.
Apply Monte Carlo sampling – a sample size of “n”
where it is suggested that n = 30.
Compute the measure of worth (PW, AW, . . )
Evaluate and draw conclusions.

26. Slide Sets to © 2005 by McGraw-Hill,19-26
Examples to StudyExamples to Study
 Examples 19.7 and 19.8 offer detailed
analyses of a simulated economic analysis
 Also, refer to the Additional Examples at the
end of the chapter
Example 19.10 – applying the normal distribution

27. Slide Sets to © 2005 by McGraw-Hill,19-27
Chapter SummaryChapter Summary
 To perform decision making under risk implies
that some parameters of an engineering
alternative are treated as random variables.
 Assumptions about the shape of the variable's
probability distribution are used to explain how the
estimates of parameter values may vary.
 Additionally, measures such as the expected
value and standard deviation describe the
characteristic shape of the distribution.

28. Slide Sets to © 2005 by McGraw-Hill,19-28
Summary - continuedSummary - continued
 In this chapter, we learned several of the simple,
but useful, discrete and continuous population
distributions used in engineering economy
-uniform and triangular - as well as specifying our
own distribution or assuming the normal
distribution.
 It is important to note that a sound
background in applied statistics is vital to the
complete understanding of the simulation
process

29. Slide Sets to © 2005 by McGraw-Hill,19-29
Chapter 19Chapter 19
End of SetEnd of Set