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Unformatted text preview: CHAPTER 13 CHAPTER PROBABILISTIC RISK PROBABILISTIC ANALYSIS ANALYSIS RANDOM VARIABLES RANDOM • Factors having probabilistic outcomes • The probability that a cost, revenue, useful life, The or other economic factor value will occur, is usually considered to be the subjectively estimated likelihood that an event (value) occurs estimated • Random variable information that is particularly Random helpful in decision making are the expected values and variances values • These values make the uncertainty associated These with each alternative more explicit RANDOM VARIABLES RANDOM • Capital letters such as X, Y, and Z are Capital used to represent random variables used • Lower-case letters (x,y,z) denote the Lower-case particular values that these variables take on in the sample space (I.e., the set of possible outcomes for each variable) variable) RANDOM VARIABLES RANDOM • When random variable X follows some When discrete probability distribution, its mass discrete function is usually indicated by p(x) and (x) its cumulative distribution function by P(x) P(x) • When X follows a continuous probability When distribution, its probability density function function and it cumulative distribution function are usually indicated by f(x) and F(x), respectively DISCRETE RANDOM VARIABLES DISCRETE • A random variable X is discrete if it can random take on a finite number of values (x1,x2… take xL) • The probability that a discrete random The variable X takes on the value xi is given by Pr{X = xi} = p(xi) for i = 1,2,….,L (i is a for sequential index of the discrete values, sequential xi, that the variable takes on) where p(xi) > 0 and Σ i p(xi) = 1 where CONTINUOUS RANDOM VARIABLES CONTINUOUS A random variable is continuous if: Pr{c < X < d} =∫cd f(x)dx f(x)dx In the nonnegative function f(x),this is the probability that In X is within the set of real numbers (c,d) is ∫-∞∞f(x)dx = 1 The probability that the value X is less than or equal x = k, The the cumulative distribution function F(x) for a continuous case is continuous Pr{X < k} = F(k) = ∫-∞k f(x)dx f(x)dx Pr{c < X < d} =∫cd f(x)dx = F(d) – F( c ) Pr{c F( F( • In most applications, continuous random variables In represent measured data, such as time, cost and revenue on a continuous scale revenue MATHEMATICAL EXPECTATIONS AND SELECTED STATISTICAL MOMENTS SELECTED • The expected value of a single random variable X, The (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 central • E(X) is the first moment of the random variable E(X) about the origin and is called the mean of the distribution distribution E(X) = Σ i xi p( xi ) for x discrete and i = 1,2,…,L E(X) E(X) = ∫-∞∞[x – E(X)]2 f(x)dx for x continuous E(X) • From binomial expansion of [X – E(X)]2 MATHEMATICAL EXPECTATIONS AND SELECTED STATISTICAL MOMENTS SELECTED V(X) = E(X2) – [E(X)]2 • V(X) is the second moment of the random V(X) variable around the origin : the expected value of X2, minus the square of its mean • V(X) is the variance of the random variable X V(X) = Σ i x2p(xi) – [E(X)]2 for x discrete V(X) V(X) = ∫-∞∞xi2(x)dx – [E(X)]2 for x continuous V(X) for • The standard deviation of a random variable, The SD(X) is the positive square root of the variance SD(X) SD(X) = [V(X)]1/2 MULTIPLICATION OF A RANDOM VARIABLE BY A CONSTANT VARIABLE • When a random variable, X, is multiplied by a When constant, c, the expected value E(cX), and the variance, V(cX) are: variance, E(cX) = cE(X) = Σ i cxi p(xi) for discrete E(cX) cx E(cX) = cE(X) = ∫-∞∞cx f(x)dx for continuous E(cX) V(cX) = E{ [cX – E(cX)]2 } =E{c2X2 – 2c2X . E(X) + c2 [E(X)]2 } E(X) [E(X)] =c2E{ [X – E(X)]2 } MULTIPLICATION OF TWO INDEPENDENT VARIABLES VARIABLES • When a random variable, Z, is a product of two When independent random variables, X and Y, the expected value, E(Z), and the variance, V(Z) are expected Z= XY E(Z) = E(X) E(Y) V(Z) = E [XY – E(X)]2 = E { X2Y2 – 2XY E(XY) + [E(XY)]2 } =EX2 EY2 – [E(X) E(Y)]2 EY 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 VARIABLES V(Z) = { V(X) + [E(X)]2 } { V(Y) + [E(Y)]2 } – [E(X)]2 [E(Y)]2 V(Y) [E(X)] [E(Y)] Or V(Z) = V(X) [E(Y)]2 + V(Y) [E(X)]2 + V(X) V(Y) EVALUATION OF PROJECTS WITH DISCRETE RANDOM VARIABLES DISCRETE • Expected value and variance concepts Expected apply theoretically to long-run conditions in which it is assumed that the event is going to occur repeatedly going • However, application of these concepts is However, often useful when investments are not going to be made repeatedly over the long run long EVALUATION OF PROJECTS WITH CONTINUOUS RANDOM VARIABLES CONTINUOUS Two Frequently Used Assumptions • Uncertain cash-flow amounts are Uncertain distributed according to the normal distribution distribution • Uncertain cash flow amounts are Uncertain statistically independent statistically – no correlation between cash flow amounts is no assumed assumed EVALUATION OF PROJECTS WITH CONTINUOUS RANDOM VARIABLES CONTINUOUS If there is a linear combination of two or more If independent cash flow amounts (i.e., PW = c0F0 + … +cNFN, where ck values are coefficients and Fk values are periodic net coefficients cash flows) the expression V(PW) reduces to cash V(PW) = Σ k=0N ck2 V(Fk) V(PW) V(F E(PW) = Σ k=0N ckE(Fk) E(PW) EVALUATION OF UNCERTAINTY USING MONTE CARLO SIMULATION USING • Computer-assisted simulation tool for Computer-assisted analyzing more complex project uncertainties uncertainties • Monte Carlo simulation generates random Monte outcomes for probabilistic factors which imitate the randomness inherent in the original problem original EVALUATION OF UNCERTAINTY USING MONTE CARLO SIMULATION MONTE • Construct an analytical model that represents the Construct actual decision situation actual • Develop a probability distribution from subjective Develop or historical data for each uncertain factor in the model model • Sample outcomes are randomly generated by Sample using probability distribution for each uncertain quantity and then used to determine a trial outcome for the model outcome • Repeating sampling process many times leads to Repeating a frequency distribution of trial outcomes, which are used to make probabilistic statements DECISION TREES DECISION • Also called decision flow networks and Also decision diagrams decision • Powerful means of depicting and facilitating Powerful analysis of important problems, especially those that involve sequential decisions and variable outcomes over time variable • Practical tool because it permits large Practical complicated problems to be reduced to a series of smaller simple problems series • Enable objective analysis and decision making Enable that includes explicit consideration of the risk and effect of the future and GENERAL PRINCIPLE OF DIAGRAMING GENERAL The Decision Tree Diagram Should Show the Following (With square symbol to depict decision node and circle (With symbol to depict chance outcome node): symbol 1. All initial or immediate alternatives among which the 1. decision maker wishes to choose decision 2. All uncertain outcomes and future alternatives the 2. decision maker wishes to consider Note alternatives at any point and outcomes at any chance outcome node must be: chance • Mutually exclusive Mutually • Collectively exhaustive; that is, one event must be Collectively chosen or something must occur if the decision point or outcome node is reached ...
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