Chapter 19 decision-making under risk

Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005

© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-1

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



LEARNING OBJECTIVESLEARNING OBJECTIVES

1. Certainty and risk

2. Variables and distributions

3. Random samples

4. Average and dispersion

5. Monte Carlo simulation



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



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



Two Ways to Consider Risk in Two Ways to Consider Risk in Decision MakingDecision Making

Expected Value (EV) analysisApplies the notion of expected value (Chapter 18)Calculation of EV of a given outcomeSelection of the outcome with the most

advantageous outcome

Simulation AnalysisForm of generating artificial data from assumed

probability distributionsRelies on the use of random variables and the laws

associated with the algebra of random variables



Sct 19.2 Elements Important to Decision Sct 19.2 Elements Important to Decision Making Under RiskMaking Under Risk

The concept of a random variableA decision rule that assigns an outcome to a

sample spaceDiscrete variable or Continuous variable

Discrete variable – finite number of outcomes possible Continuous variable – infinite number of outcomes

ProbabilityNumber between 0 and 1Expresses the “chance” in decimal form that a

random variable will take on any specific value



Types of Random VariablesTypes of Random Variables

Continuous

Discrete



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)



Three Common Random VariablesThree Common Random Variables

Uniform – equally likely outcomes

Triangular

Normal

Study

Example 19.3



Discrete Density and Cumulative ExampleDiscrete Density and Cumulative Example

pdf cdf



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 … orContinuous distributions



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



Sct 19.4 Expected Value and Standard Sct 19.4 Expected Value and Standard DeviationDeviation

Two important parameters of a given random variable:Mean -

Measure of central tendency

Standard Deviation - Measure of variability or spread

Two Concepts to work withinPopulationSample from a population



Sample

Population vs SamplePopulation vs Sample

Population - population mean 2 - population

variance - population

standard deviation Often sample from a

population in order to make estimates

X

2s

s

Sample mean

Sample variance

Sample standard deviation

These values, properly sampled, attempt to estimate their population counterparts



Important RelationshipsImportant Relationships

Population Mean 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



Variance and Standard DeviationVariance and Standard Deviation

Notes relating to variance and standard deviation properties

Illustration of variances for discrete and continuous distributions



Population vs. SamplePopulation vs. Sample

Variance of a population

Standard deviation of a population:

N2

ii=1

(x )

N

2 Var(X)X X

N2

i2 i=1

(x )

N

Variance of a sample

Standard deviation of a sample

2

2 2

1 1iX n

s Xn n

S is termed an unbiased estimator of the population standard deviation



Combining the Average and Standard Combining 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



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 15 210

10.2 $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



Sct 19.5 Monte Carlo Sampling and Sct 19.5 Monte Carlo Sampling and Simulation AnalysisSimulation Analysis

Simulation involves the generation of artificial data from a modeled system

Monte Carlo SamplingThe generation of samples of size n for selected

parameters of formulated alternativesThe sampled parameters are expected to vary

according to a stated probability distribution (assumed)



Key Assumption - IndependenceKey Assumption - Independence For a given problem:

All parameters are assumed to be independentOne variable’s distribution in no way impacts any

other variable’s distributionTermed:

Property of independent random variables

The modeling approach follows a 7-step process (page 678)



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.



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.



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



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.



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



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.



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



Chapter 19Chapter 19End of SetEnd of Set

Chapter 19 decision-making under risk

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