Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,Slide Sets to accompany Blank&Tarquin,Engineering Economy,6,th,Edition,2005,2005 by McGraw-Hill,New York,N.Y All Rights Reserved,19-,*,Click to edit Master title style,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 OBJECTIVES,Certainty and risk,Variables and distributions,Random samples,Average and dispersion,Monte Carlo simulation,Sct 19.1 Interpretation of Certainty,Risk,and 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 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 Decision 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,Sct 19.2 Elements Important to Decision Making 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,Types of Random Variables,Continuous,Discrete,Distributions-Continuous Variables,Probability Distribution(pdf),A function that describes how probability is distributed over the different values of a variable,P(X,i,),=probability that X=X,i,Cumulative Distribution(cdf),Accumulation of probability over all values of a variable up to and including a specified value,F(X,i,),=sum of all probabilities through the value X,i,=P(X,X,i,),Three Common Random Variables,Uniform equally likely outcomes,Triangular,Normal,Study,Example 19.3,Discrete Density and Cumulative Example,pdf cdf,Sct 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,Sampling 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 Deviation,Two important parameters of a given random variable:,Mean-,Measure of central tendency,Standard Deviation-,Measure of variability or spread,Two Concepts to work within,Population,Sample from a population,Sample,Population vs Sample,Population,-population mean,2,-population variance,-,population standard deviation,Often sample from a population in order to make estimates,Sample mean,Sample variance,Sample standard deviation,These values,properly sampled,attempt to estimate their population counterparts,Important 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,Variance and Standard Deviation,Notes relating to variance and standard deviation properties,Illustration of variances for discrete and continuous distributions,Population vs.Sample,Variance of a population,Standard deviation of a population:,Variance of a sample,Standard deviation of a sample,S is termed an,unbiased estimator,of the population standard deviation,Combining the Average and Standard Deviation,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,Con