CHAPTER 13 PROBABILISTIC RISK ANALYSIS RANDOM VARIABLES Factors having probabilistic outcomesFactors having probabilistic outcomes The probability that.

Slides:



Advertisements
Similar presentations
Chapter 4 Probability and Probability Distributions
Advertisements

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 4-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter.
April 2, 2015Applied Discrete Mathematics Week 8: Advanced Counting 1 Random Variables In some experiments, we would like to assign a numerical value to.
Copyright ©2012 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved. Engineering Economy, Fifteenth Edition By William.
Engineering Economic Analysis Canadian Edition
Engineering Economics in Canada Chapter 12 Dealing with Risk: Probability Analysis.
FREQUENCY ANALYSIS Basic Problem: To relate the magnitude of extreme events to their frequency of occurrence through the use of probability distributions.
Review of Basic Probability and Statistics
Discrete Random Variables and Probability Distributions
BCOR 1020 Business Statistics Lecture 9 – February 14, 2008.
Probability and Statistics Review
4. Review of Basic Probability and Statistics
CHAPTER 6 Statistical Analysis of Experimental Data
2. Random variables  Introduction  Distribution of a random variable  Distribution function properties  Discrete random variables  Point mass  Discrete.
3-1 Introduction Experiment Random Random experiment.
Chapter 16: Random Variables
CEEN-2131 Business Statistics: A Decision-Making Approach CEEN-2130/31/32 Using Probability and Probability Distributions.
Random Variable and Probability Distribution
Modern Navigation Thomas Herring
Chapter 21 Random Variables Discrete: Bernoulli, Binomial, Geometric, Poisson Continuous: Uniform, Exponential, Gamma, Normal Expectation & Variance, Joint.
Chapter 14 Risk and Uncertainty Managerial Economics: Economic Tools for Today’s Decision Makers, 4/e By Paul Keat and Philip Young.
Probability and Probability Distributions
Problem A newly married couple plans to have four children and would like to have three girls and a boy. What are the chances (probability) their desire.
Chapter 7: Random Variables
QA in Finance/ Ch 3 Probability in Finance Probability.
Engineering Economy, Sixteenth Edition Sullivan | Wicks | Koelling Copyright ©2015, 2012, 2009 by Pearson Education, Inc. All rights reserved. TABLE 12-1.
Chapter 12 Review of Calculus and Probability
Chapter 5 Discrete Random Variables and Probability Distributions ©
Chapter 8 Probability Section R Review. 2 Barnett/Ziegler/Byleen Finite Mathematics 12e Review for Chapter 8 Important Terms, Symbols, Concepts  8.1.
Chapter © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or.
Chap 9 Estimating Volatility : Consolidated Approach.
Dr. Gary Blau, Sean HanMonday, Aug 13, 2007 Statistical Design of Experiments SECTION I Probability Theory Review.
Theory of Probability Statistics for Business and Economics.
Day 2 Review Chapters 5 – 7 Probability, Random Variables, Sampling Distributions.
1 Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems.
Using Probability and Discrete Probability Distributions
Uncertainty in Future Events Chapter 10: Newnan, Eschenbach, and Lavelle Dr. Hurley’s AGB 555 Course.
1 G Lect 3b G Lecture 3b Why are means and variances so useful? Recap of random variables and expectations with examples Further consideration.
CPSC 531: Probability Review1 CPSC 531:Probability & Statistics: Review II Instructor: Anirban Mahanti Office: ICT 745
Review of Probability Concepts ECON 6002 Econometrics Memorial University of Newfoundland Adapted from Vera Tabakova’s notes.
Engineering Economic Analysis Canadian Edition
Random Variables. A random variable X is a real valued function defined on the sample space, X : S  R. The set { s  S : X ( s )  [ a, b ] is an event}.
BINOMIALDISTRIBUTION AND ITS APPLICATION. Binomial Distribution  The binomial probability density function –f(x) = n C x p x q n-x for x=0,1,2,3…,n for.
Discrete Random Variables A random variable is a function that assigns a numerical value to each simple event in a sample space. Range – the set of real.
Probability & Statistics I IE 254 Summer 1999 Chapter 4  Continuous Random Variables  What is the difference between a discrete & a continuous R.V.?
Chap 4-1 A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc. A Course In Business Statistics 4 th Edition Chapter 4 Using Probability and Probability.
Discrete Random Variables. Numerical Outcomes Consider associating a numerical value with each sample point in a sample space. (1,1) (1,2) (1,3) (1,4)
Outline of Chapter 9: Using Simulation to Solve Decision Problems Real world decisions are often too complex to be analyzed effectively using influence.
Basic Numerical Procedures Chapter 19 1 Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008.
1 G Lect 2M Examples of Correlation Random variables and manipulated variables Thinking about joint distributions Thinking about marginal distributions:
Stats Probability Theory Summary. The sample Space, S The sample space, S, for a random phenomena is the set of all possible outcomes.
Exam 2: Rules Section 2.1 Bring a cheat sheet. One page 2 sides. Bring a calculator. Bring your book to use the tables in the back.
Review of Probability. Important Topics 1 Random Variables and Probability Distributions 2 Expected Values, Mean, and Variance 3 Two Random Variables.
1 1 Slide Simulation Professor Ahmadi. 2 2 Slide Simulation Chapter Outline n Computer Simulation n Simulation Modeling n Random Variables and Pseudo-Random.
Random Variables. Numerical Outcomes Consider associating a numerical value with each sample point in a sample space. (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Probabilistic Cash Flow Analysis.
Chapter 2: Probability. Section 2.1: Basic Ideas Definition: An experiment is a process that results in an outcome that cannot be predicted in advance.
Chapter 8: Probability: The Mathematics of Chance Probability Models and Rules 1 Probability Theory  The mathematical description of randomness.  Companies.
1 Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems.
Chap 4-1 Chapter 4 Using Probability and Probability Distributions.
Evaluating Hypotheses. Outline Empirically evaluating the accuracy of hypotheses is fundamental to machine learning – How well does this estimate accuracy.
Probabilistic Cash Flow Analysis
MECH 373 Instrumentation and Measurements
CHAPTER 2 RANDOM VARIABLES.
Chapter 5 Statistical Models in Simulation
Chapter 12: Probabilistic Risk Analysis
Probability.
AP Statistics Chapter 16 Notes.
Discrete Random Variables and Probability Distributions
Chapter 12: Probabilistic Risk Analysis
Presentation transcript:

CHAPTER 13 PROBABILISTIC RISK ANALYSIS

RANDOM VARIABLES Factors having probabilistic outcomesFactors having probabilistic outcomes The probability that a cost, revenue, useful life, or other economic factor value will occur, is usually considered to be the subjectively estimated likelihood that an event (value) occursThe probability that a cost, revenue, useful life, or other economic factor value will occur, is usually considered to be the subjectively estimated likelihood that an event (value) occurs Random variable information that is particularly helpful in decision making are the expected values and variancesRandom variable information that is particularly helpful in decision making are the expected values and variances These values make the uncertainty associated with each alternative more explicitThese values make the uncertainty associated with each alternative more explicit

RANDOM VARIABLES Capital letters such as X, Y, and Z are used to represent random variablesCapital letters such as X, Y, and Z are used to represent random variables Lower-case letters (x,y,z) denote the particular values that these variables take on in the sample space (I.e., the set of possible outcomes for each variable)Lower-case letters (x,y,z) denote the particular values that these variables take on in the sample space (I.e., the set of possible outcomes for each variable)

RANDOM VARIABLES When random variable X follows some discrete probability distribution, its mass function is usually indicated by p(x) and its cumulative distribution function by P(x)When random variable X follows some discrete probability distribution, its mass function is usually indicated by p(x) and its cumulative distribution function by P(x) When X follows a continuous probability distribution, its probability density function function and it cumulative distribution function are usually indicated by f(x) and F(x), respectivelyWhen X follows a continuous probability distribution, its probability density function function and it cumulative distribution function are usually indicated by f(x) and F(x), respectively

DISCRETE RANDOM VARIABLES A random variable X is discrete if it can take on a finite number of values (x 1,x 2 …x L )A random variable X is discrete if it can take on a finite number of values (x 1,x 2 …x L ) The probability that a discrete random variable X takes on the value x i is given byThe probability that a discrete random variable X takes on the value x i is given by Pr{X = x i } = p(x i ) for i = 1,2,….,L (i is a sequential index of the discrete values, x i, that the variable takes on) where p(x i ) > 0 and  i p(x i ) = 1

CONTINUOUS RANDOM VARIABLES A random variable is continuous if: Pr{c < X < d} =∫ c d f(x)dx In the nonnegative function f(x),this is the probability that X is within the set of real numbers (c,d) ∫ -∞ ∞ f(x)dx = 1 The probability that the value X is less than or equal x = k, the cumulative distribution function F(x) for a continuous case is Pr{X < k} = F(k) = ∫ -∞ k f(x)dx Pr{c < X < d} =∫ c d f(x)dx = F(d) – F( c ) In most applications, continuous random variables represent measured data, such as time, cost and revenue on a continuous scaleIn most applications, continuous random variables represent measured data, such as time, cost and revenue on a continuous scale

MATHEMATICAL EXPECTATIONS AND SELECTED STATISTICAL MOMENTS The expected value of a single random variable X, (E(X), is a weighted average of the distributed values x that it takes on and is a measure of the central location of the distributionThe expected value of a single random variable X, (E(X), is a weighted average of the distributed values x that it takes on and is a measure of the central location of the distribution E(X) is the first moment of the random variable about the origin and is called the mean of the distributionE(X) is the first moment of the random variable about the origin and is called the mean of the distribution E(X) =  i x i p( x i ) for x discrete and i = 1,2,…,L E(X) = ∫ -∞ ∞ [x – E(X)] 2 f(x)dx for x continuous

MATHEMATICAL EXPECTATIONS AND SELECTED STATISTICAL MOMENTS FromFrom binomial expansion of [X – E(X)] 2 V(X) = E(X 2 ) E(X 2 ) – [E(X)] 2 V(X)V(X) is the second moment of the random variable around the origin : the expected value of X 2, X 2, minus the square of its mean V(X)V(X) is the variance of the random variable X V(X) = i i i i x 2 p(x i ) x 2 p(x i ) – [E(X)] 2 [E(X)] 2 for x discrete V(X) = ∫ -∞ ∞ x i 2 (x)dx ∫ -∞ ∞ x i 2 (x)dx – [E(X)] 2 [E(X)] 2 for x continuous TheThe standard deviation of a random variable, SD(X) is the positive square root of the variance SD(X) = [V(X)] 1/2

MULTIPLICATION OF A RANDOM VARIABLE BY A CONSTANT WhenWhen a random variable, X, is multiplied by a constant, c, the expected value E(cX), and the variance, V(cX) are: E(cX) = cE(X) = i i i i cx i cx i p(x i ) p(x i ) for discrete E(cX) = cE(X) = ∫ -∞ ∞ cx ∫ -∞ ∞ cx f(x)dx f(x)dx for continuous V(cX) = E{ [cX – E(cX)] 2 E(cX)] 2 } =E{c 2 X 2 =E{c 2 X 2 – 2c 2 X 2c 2 X.E(X) + c 2 c 2 [E(X)] 2 [E(X)] 2 } =c 2 E{ =c 2 E{ [X – E(X)] 2 E(X)] 2 }

MULTIPLICATION OF TWO INDEPENDENT VARIABLES When a random variable, Z, is a product of two independent random variables, X and Y, the expected value, E(Z), and the variance, V(Z) areWhen a random variable, Z, is a product of two independent random variables, X and Y, the expected value, E(Z), and the variance, V(Z) are Z= XY E(Z) = E(X) E(Y) V(Z) = E [XY – E(X)] 2 = E { X 2 Y 2 – 2XY E(XY) + [E(XY)] 2 } =EX 2 EY 2 – [E(X) E(Y)] 2 But the variance of any random variable, V(RV), is V(RV) = E[(RV) 2 ] – [E(RV)] 2 E[(RV) 2 ] = V(RV) + [E(RV)] 2

MULTIPLICATION OF TWO INDEPENDENT VARIABLES V(Z) = { V(X) + [E(X)] 2 [E(X)] 2 } { V(Y) + [E(Y)] 2 [E(Y)] 2 } – [E(X)] 2 [E(X)] 2 [E(Y)] 2 Or V(Z) = V(X) [E(Y)] 2 [E(Y)] 2 + V(Y) [E(X)] 2 [E(X)] 2 + V(X) V(Y)

EVALUATION OF PROJECTS WITH DISCRETE RANDOM VARIABLES Expected value and variance concepts apply theoretically to long-run conditions in which it is assumed that the event is going to occur repeatedlyExpected value and variance concepts apply theoretically to long-run conditions in which it is assumed that the event is going to occur repeatedly However, application of these concepts is often useful when investments are not going to be made repeatedly over the long runHowever, application of these concepts is often useful when investments are not going to be made repeatedly over the long run

EVALUATION OF PROJECTS WITH CONTINUOUS RANDOM VARIABLES Two Frequently Used Assumptions Uncertain cash-flow amounts are distributed according to the normal distributionUncertain cash-flow amounts are distributed according to the normal distribution Uncertain cash flow amounts are statistically independentUncertain cash flow amounts are statistically independent –no correlation between cash flow amounts is assumed

EVALUATION OF PROJECTS WITH CONTINUOUS RANDOM VARIABLES If there is a linear combination of two or more independent cash flow amounts (i.e., PW = c 0 F 0 + … +c N F N, where c k values are coefficients and F k values are periodic net cash flows) the expression V(PW) reduces to V(PW) =  k=0 N c k 2 V(F k ) E(PW) =  k=0 N c k E(F k )

EVALUATION OF UNCERTAINTY USING MONTE CARLO SIMULATION Computer-assisted simulation tool for analyzing more complex project uncertaintiesComputer-assisted simulation tool for analyzing more complex project uncertainties Monte Carlo simulation generates random outcomes for probabilistic factors which imitate the randomness inherent in the original problemMonte Carlo simulation generates random outcomes for probabilistic factors which imitate the randomness inherent in the original problem

EVALUATION OF UNCERTAINTY USING MONTE CARLO SIMULATION Construct an analytical model that represents the actual decision situationConstruct an analytical model that represents the actual decision situation Develop a probability distribution from subjective or historical data for each uncertain factor in the modelDevelop a probability distribution from subjective or historical data for each uncertain factor in the model Sample outcomes are randomly generated by using probability distribution for each uncertain quantity and then used to determine a trial outcome for the modelSample outcomes are randomly generated by using probability distribution for each uncertain quantity and then used to determine a trial outcome for the model Repeating sampling process many times leads to a frequency distribution of trial outcomes, which are used to make probabilistic statementsRepeating sampling process many times leads to a frequency distribution of trial outcomes, which are used to make probabilistic statements

DECISION TREES Also called decision flow networks and decision diagramsAlso called decision flow networks and decision diagrams Powerful means of depicting and facilitating analysis of important problems, especially those that involve sequential decisions and variable outcomes over timePowerful means of depicting and facilitating analysis of important problems, especially those that involve sequential decisions and variable outcomes over time Practical tool because it permits large complicated problems to be reduced to a series of smaller simple problemsPractical tool because it permits large complicated problems to be reduced to a series of smaller simple problems Enable objective analysis and decision making that includes explicit consideration of the risk and effect of the futureEnable objective analysis and decision making that includes explicit consideration of the risk and effect of the future

GENERAL PRINCIPLE OF DIAGRAMING The Decision Tree Diagram Should Show the Following (With square symbol to depict decision node and circle symbol to depict chance outcome node): 1. All initial or immediate alternatives among which the decision maker wishes to choose 2. All uncertain outcomes and future alternatives the decision maker wishes to consider Note alternatives at any point and outcomes at any chance outcome node must be: Mutually exclusiveMutually exclusive Collectively exhaustive; that is, one event must be chosen or something must occur if the decision point or outcome node is reachedCollectively exhaustive; that is, one event must be chosen or something must occur if the decision point or outcome node is reached