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# Class Note for EMSE 269 with Professor Dorp at GW

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Date Created: 02/07/15

EMSE 269 Elements of Problem Solving and Decision Making 5103 7 PROBABILITY BASICS A an event with possible outcomes QZUA39 Total Event i1 Example A Flipping a Coin A1 2 Heads A2 Tails Q 2 Heads or Tails Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter 7 Page 69 EMSE 269 Elements of Problem Solving and Decision Making 5103 PROBABILITY CALCULUS Probability Rules may be derived using VENN DIAGRAMS 1 Probabilities must lie between 0 and 1 for all possible outcomes in omega 0 g PrA1 sly11 c Q Q Ratio of area of the event and the area of the total rectangle can be interpreted as the probability of the event Instructor Dr J Rene van Dorp Chapter7 Page 70 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 2 Probabilities must add up if not both events can occur at the same time Ala12 21 gt PI A1UA2 PrA1PrA2 Instructor Dr J Rene van Dorp Chapter7 Page 71 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 3 If A 31477 are all the possible outcomes and not two of these can occur at the same time their Total Probability must equal 1 4 mA vz Mugr r2 mug11 1 Q 45 ND win A 3A are said to be Collectively Exhaustive and Mutually Exclusive Instructor Dr J Rene van Dorp Chapter7 Page 72 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 4 The probability of the complement of A1 equals 1 minus the probability of A1 PM 1 PrltA1gt Eel Instructor Dr J Rene van Dorp Chapter7 Page 73 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 5 If two events can occur at the same time the probability of either of them happening or both equals the sum of their individual probability minus the probability of them both happening at the same time P1 A1 U A2 PrA1 PrA2 P1 A1 A2 Instructor Dr J Rene van Dorp Chapter7 Page 74 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 6 Conditional Probability Intuition If I know that the Market as a whole will go up the Chances of thestock of an individual company going up will increase Dow Jones Up i L 539 Stock PriCe Up New Total Event based on condition that we know that Dow Jones went up Q Instructor Dr J Rene van Dorp Chapter7 Page 75 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 PrStock T Dow T PrD0w T PrStock Tl Dow T Informally Conditioning on an event coincides with reducing the total event to the conditioning event PrA NB PrA1 Bl 1 1 Thus PrBl Example The probability of drawing an ace of spades in a deck of 52 cards equals 152 However if I tell you that l have an Ace in my hands the probability of it being the Ace of Spades equal Instructor Dr J Rene van Dorp Chapter7 Page 76 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 PrAce Space 152 1 PrAce 452 4 PrSpadeS Ace Pr A GB PrBl A1 1 1 Also PI39A1 7 Multiplicative Rule Calculating the probability of two events happening at the same time 131041 Bl PrBl I A1 PrA1 PrA1BlPrBI Instructor Dr J Rene van Dorp Chapter7 Page 77 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 0 Independence between two events lnformally two events are independents if information about one does not provide you any information about the other 1 EventA with possible outcomes Aquot 39 3AM 2 Event B with possible outcomes 3939 39 33m For Example A is the event of Flipping a Coin and B is the event of throwing a dice If you know the outcome of flipping the coin you do not learn anything about the outcome of throwing the dice Hence these two events are independent Instructor Dr J Rene van Dorp Chapter7 Page 78 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Formal definition of independence between two eventA and Event B 1 ltgt PrAl B PrAl VA B 2 ltgt PrB Ai PrBJVAl B 3 ltgt PrAl NB 2 PrAl PrBj v AlBj lnformally Any information about A does not tell me anything about B and vice versa Instructor Dr J Rene van Dorp Chapter7 Page 79 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Independence in Influence Diagrams o No arrow between two chance nodes implies independence between the uncertain events 0 Ah arrow from a chance event A to a chance event B does not mean that quotA causes Bquot It indicates that information about A helps in determining the likelihess of outcomes of B o Conditional Independence The performance of a person on any IQ test is uncertain and may range anywhere from 0 to 100 However were you to know that the person is highly intelligent hisher score will be high eg ranging anywhere from 90 to 100 The person s IQ does not explain the remaining uncertainty but for example other external conditions do such as eg a good night sleep the previous night On any two IQ tests these remaining uncertainties may be Instructor Dr J Rene van Dorp Chapter7 Page 80 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 reasonably modeled as independent were we to know the IQ of the person involved Formal definition Event A and Event B are conditionally independent given event C with possible outcomes Gquot 39 quotsz 139 C PrAi IBJ DCk PrAi ICk9VAiaBj9Ck 2 lt3 PrB AlCk PrB ick VA BCk OF Instructor Dr J Rene van Dorp Chapter7 Page 81 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 3 em11 m3 Ck PrAl CkgtlltPrBj CkVAlBjCk Informally If I already know C any information or knowledge about A does not tell me anything more about B and vice versa Conditional Independence in Influence Diagrams Instructor Dr J Rene van Dorp Chapter7 Page 82 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 0 Law of Total Probability Let g 3 be mutually exclusive collectively exhaustive PrA1 PrA1 m B1 PrA1 m Bz PrA1 m B3 lt3 PrA1 PrA1 31 PrBl PrA1 32 Pr32 PrA1 33 Pr33 B1 B3 5 Bz i Instructor Dr J Rene van Dorp Chapter7 Page 83 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Example Law of Total Probability SYSTEM X Xfailure No Failure A Component A fails B Component B fails C Component C fails Assume that components A B and C operate independently Instructor Dr J Rene van Dorp Chapter7 Page 84 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 TASK Write the probability of failure PrX as a function of the component failure probabilities PrA PrB and PrC 1PrX PrXl APrA PrXlZPrZ1 PrA PrXl Emu 2PrXlA7 PrXlBA7PrB l ZPrXiEZPrElZ PrX l BZPrBOPr PrXlBZPrB Substituting the result of 2 into 1 yields 3PrX PrA PrX B Z PrB PrZ Hence we need to further develop PrX l 314 Instructor Dr J Rene van Dorp Chapter7 Page 85 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 4PrXiBA7 PrX l CBA7PrC l BZPrX l BZPr lBZ PrX i112 1PrC 0Pr PrC Substituting the result of 4 into 3 yields 5 PrX PrA PrCPrBPrA7 Finally using the complement rule PrZ 1 PrA and substituting it in 5 yields Pr X PrA PrC PrB PrCPrBPrA Instructor Dr J Rene van Dorp Chapter7 Page 86 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Example Oil Wildcatter Problem Max Profit Dry 100K Drill at Site 1 K Y Low 150K High g 500K Dry 02 200K Drill at Site 2 50K Payoff at site 1 is uncertain Dominating factor in eventual payoff at Site 1 is the presence of a dome or not Instructor Dr J Rene van Dorp Chapter7 Page 87 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 l I DOME gt I x I x I I i l PrDome PrNo Dome l 7 0600 0400 l Outcome PrOutcomeIDome Outcome PrOutcomeINo Dome Dw 0600 Dry 0850 Low 0250 Low 0125 Hiqh 0150 Hiqh 0025 Instructor Dr J Rene van Dorp Chapter7 Page 88 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 0 Law of Total Probability PrDry PrDry l Dome PrD0me PrDry l NODome PrNOD0me PrDry 0600 0600 0850 0400 0700 PrL0w PrL0w l Dome PrD0me PrL0w l NOD0me Pr NOD0me PrL0w 0250 0600 0125 0400 0200 PrHigh PrHigh l D0mePrD0me PrHigh l NOD0mePr NOD0me PrL0w 0150 0600 0025 0400 0100 Instructor Dr J Rene van Dorp Chapter 7 Page 89 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 D 0600 W 100K Dome 0600 Low 0250 150K High 0150 500K Dry 0850 1OOK No Dome 0400 Low 0125 150K High 0025 500K LAW OF TOTAL PROBABILITY Dry 0600 0600 0850 0400 070 100K Low 0250 0600 0125 0400 020 150K 39 h 0150 0600 0025 0400 010 500K INFORMALLY when we apply LOTP we are collapsing a probability tree Instructor Dr J Rene van Dorp Chapter7 Page 90 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Bayes Theorem Let 3 3 be mutually exclusive collectively exhaustive Mm Instructor Dr J Rene van Dorp Chapter7 Page 91 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 From the multiplicative rule follows that 1PrA1 Bj PrBj l A1PrA1 PrA1lBjPrBj Dividing the LHS and RHS by PrA1 yields Pr A B Pr B 2pr3jiAlM PrA1 We may rewrite PrA1 using the Law of Total Probability yielding 3PrA1 PrA1 iBlgtPrltBlgtPrltA1 iBzgtPrltBZgtPrltA1 lB3PrB3 Substituting the result of 3 into 2 gives perhaps the most well known theorem in probability theory Bayes Theorem Instructor Dr J Rene van Dorp Chapter7 Page 92 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE mar Elements m Pvublem Sulvmg and Declsmn Making 5mm PIA1lBJPrB 4PrBJlA1 PIA1lBlPrBlPr11lePrBzPrA1lBgPrBg Thomas Bayes lived from 1702 to 1761 Bayes set out his theory of probability in 1764 His conclusions were accepted by Laplace in 1781 rediscovered by Condorcet and remained unchallenged until Boole questioned them Since then Bayes39 techniques have been subject to controversy httpWWW andcsstandacukNhistorVMathematiciansBayeshtml lns1mcluv D J Renevan Duvp Chaple r Page 53 Suuvce MaklndHavdDeclsmns AnlmvuducllunluDeclslunAnalvsls mm Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Oil Wildcatter Problem Example Continued 0 Bayes Theorem We drilled at site 1 and the well is a high producer Given this new information what are the Chances that a dome exists Perhaps that information is important when attracting additional investors PrHighD0mePrD0me 1 PrD0me Hzgh Hazy 2 PrHl39gh PrHl39gh D0mePrD0me PrHl39gh NOD0mePrNOD0me Instructor Dr J Rene van Dorp Chapter7 Page 94 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 PrHigh D0me Pr Dame 3 PrDome I PrHighD0mePrDomePrHighNOD0me PrNoDome 4 PrDome High 2 0150039600 090 O150O60000250O4OO PrDome Prior Probability PrDomeData Posterior Probability Instructor Dr J Rene van Dorp Chapter7 Page 95 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Dome 0600 M Low 0250 150K High 0150 Dry 0850 100K No Dome 0400f Low 0125 150K Hi h 0025 500K BAYES THEOREM Dome 100K Dry 07 No Dome 100K 150K Low 02 No Dome 150K Dome 090 500K High 010 No Dome 500K Instructor Dr J Rene van Dorp Chapter7 Page 96 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 When we reverse the order of Chance nodes in a decision tree we need to apply Bayes Theorem o Calculating posterior probabilities using a Table PrDome PrNo Dome 0600 0400 X PrXDome PrXNo Dome PrX n Dome PrX n No Dome PrX PrDomelX PrNo DomeIX Check Dry 0600 0850 0360 0340 0700 0514 0486 1000 Low 0250 0125 0150 0050 0200 0750 0250 1000 High 0150 0025 0090 0010 0100 0900 0100 1000 Check 1000 1000 0600 0400 1000 Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter 7 Page 97 EMSE 269 Elements of Problem Solving and Decision Making 5103 AFTER BAYES THEOREM Dome 0514 100K Dry 07 No Dome 0486 100K Dome 0750 150K Low 02 No Dome 0250 150K Dome 0900 500K High 010 No Dome 0100 500K Instructor Dr J Rene van Dorp Chapter7 Page 98 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Another Example of Bayes Theorem Game Show Suppose we have a game show host and you There are three doors and one of them contains a prize The game show host knows the door containing the prize but of course does not convey this information to you He asks you to pick a door You picked door 1 and are walking up to door 1 to open it when the game show host screams STOP You stop and the game show host shows door 3 which appears to be empty Next the game show asks quotDO YOU WANT TO SWITCH TO DOOR 2quot WHAT SHOULD YOU DO Instructor Dr J Rene van Dorp Chapter7 Page 99 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Assumption 1 The game show host will never show the door with the prize Assumption 2 The game show will never show the door that you picked o DiPrize is behind door i i13 o HiHost shows door i containing no prize after you selected Door 1 i1 3 1 Initially it seems reasonable to set PrDi 2 E Apply Law of Total Probability 3 1 1 1 1 1 PrII Prl I D MD 10gt1lt 133l23 3 32 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Use calculation rule for conditional probability and substitute the result of 1 1 1 gtllt PrH DPrD 1 PrD1H3 3mg 12213 2 3 2 Next apply the complement rule applied to a conditional probability 1 2 3 PrD2 IH31PID1 IH31 So YES you should SWITCH as you would increase your chances of winning Instructor Dr J Rene van Dorp Chapter7 Page 101 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 UNCERTAIN QUANTITIES amp RANDOM VARIABLES Event A with possible outcomes Alquot 39 3AM A A Number of Raisins in an oatmeal cookie i Raisins in an oatmeal cookie i12 n n mu 14 Total Event or Sample Space A Random Variable Y Uncertain Quantity is a function from 9 Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter 7 Page 102 EMSE 269 Elements of Problem Solving and Decision Making 5103 Define Random Variable Y The number of Raisins in a oatmeal cookie Then YAy i Outcome Al Number Associated with Outcome i Note this is in principal similar to eg a function from fZR R where f x x2 Argument X Number associated with argument X Instructor Dr J Rene van Dorp Chapter7 Page 103 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 The only difference is that the outcomes happen with a certain probability ie the outcomes are random and therefore the values of the variable 2i occur with the same probabilities The values of the variable Y occur at random and is therefore called a random variable To complicate matters we typically omit the argument ie we write Y yl From this notation it is not clear anymore that a random variable is a function from the outcome space to the real line Instructor Dr J Rene van Dorp Chapter7 Page 104 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 When number of outcomes of the event A is finite Y is a discrete random variable Discrete Probability Distribution DPD o The collection of probabilities associated with each possible outcome of Yis called the discrete probability distribution Thus if we denote PrCYZM pl Discrete probability distribution of Y ppum 1 2 0 Note gpl p O Other common notation fYyipj9i19 an fYy09y yi9i219 n9n Instructor Dr J Rene van Dorp Chapter7 Page 105 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Discrete Probability Distribution 04 03 PrYy 02 01 Discrete Probability Distribution El PFYY 01 015 03 035 01 Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter 7 Page 106 EMSE 269 Elements of Problem Solving and Decision Making 5103 Cumulative Probability Distribution CDF FY y PrY g y 100 080 055 025 010 Cumulative Distribution Function 090 085 080 075 070 065 060 PrYlty 050 045 040 035 030 025 020 015 010 005 000 Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter 7 Page 107 EMSE 269 Elements of Problem Solving and Decision Making 5103 In Decision Analysis a CDF is referred to as a CUMMULATIVE RISK PROFILE and a DPD is referred to as a RISK PROFILE Expected Value of Y also referred to as Mean Value of Y We know the random variable Y has many possible outcomes However if you were forced to give a BEST GUESS for Y what number would you give Managers CEO s Senators etc typically like POINT ESTIMATES unfortunately Why not use the average value of Y EYY Zyi PrY yr Zyi n i1 i1 Instructor Dr J Rene van Dorp Chapter7 Page 108 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 1 Raisin 010 1 1 2 Raisins 015 320 2 2 3 Raisins 030 3 3 4 Raisins 035 4 4 5 Raisins 010 5 5 Interpretation 5103 Raisins RaisinsPrYRaisins 1O10O1O 2O15O3O 3O30O9O 4O3514O 5O10O50 320 o If you were able to observe many outcomes of Y the calculated average of all the outcomes would be close to EY Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter 7 Page 109 EMSE 269 Elements of Problem Solving and Decision Making 5103 Calculation Rules for Expected Values 1 Let Z be a function of Y ie ZgY As Y is a random variable Z is a random variable and n n EZZ Zgy PrY M 2gyi n i1 i1 2 Only when the function Z is a linear function of Y the following holds gY aYbgtEZZaEYYI9 3 Let X Y be two random variables and ZXY then EZZEXXEYY Instructor Dr J Rene van Dorp Chapter7 Page 110 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Variance and Standard Deviation of Y We know the random variable Y has many possible outcomes If you were forced to give a BEST GUESS for the uncertainty in Y what number would you give Some people prefer to give the range of the outcomes of Y La the MAX VALUE of Y minus the MIN VALUE of Y However this completely ignores that some values of Y may be more likely than others SUGGESTION Calcute the BEST GUESS for the SQUARED DISTANCE from the MEAN Variance ValY 7 Instructor Dr J Rene van Dorp Chapter7 Page 111 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 VarY a EY EY2 EY2 2 Y EY E2Y EY2 2 EY EY E2Y EY2 E2Y 2 Standard Deviation CY CY Interpretation Standard deviation is the best guess distance from the mean for an arbritrary outcome and is measured in the same units as the Mean Value Instructor Dr J Rene van Dorp Chapter7 Page 112 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Calculation Rules 1 Let Z Y be random variables such that ZgY 2 gY aYb gtVarZ 2a VarY 2 Let X i1 n be a collection of independent random variables Yzixai Xl 191gtVarY 1lt li2 141709 Instructor Dr J Rene van Dorp Chapter7 Page 113 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Example 024 3575 A N 047 029 025 3575 B 035 040 Max Profit 20 35 50 9 0 95 Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter7 Page 114 EMSE 269 Elements of Problem Solving and Decision Making 5103 Alternative A Prob Profit Profitquot2 ProbProfit ProbProfitA2 Variance St Dev 024 20 400 480 9600 047 35 1225 1645 57575 029 50 2500 1450 72500 EIYlt 3575 I12780625 I 139675 11869 I1089438 E2Y EY2 EY2 E2YV 0y Alternative B Prob Profit Profitquot2 ProbProfit ProbProfitA2 Variance St Dev 025 9 81 225 2025 035 0 0 000 000 04 95 9025 3800 361000 3575 12780625 363025 235219 4849936 Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter7 Page 115 EMSE 269 Elements of Problem Solving and Decision Making 5103 Notes 0 B has high possible yield but also high risk PrPro t S 0 A 0PrPro t S 0 B 06 Example Oil Wildcatter Problem Continued 10K Drill at Site 2 Dry 07 Low 02 Drill at Site 2 High 01 ow 02 Max Pro t 100K 150K 500K 200K 50K Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter7 Page 116 EMSE 269 Elements of Problem Solving and Decision Making 5103 Drill at Site 1 Prob Profit Profitquot2 ProbProfit ProbProfitAZ Variance St Dev 07 100 10000 7000 700000 02 1 50 22500 3000 450000 01 500 250000 5000 2500000 1ooo gt E Y I 100 I 3650000 I3640000 1907878 E2m EY2 l oy EY2 E2Y Drill at Site 2 Prob Profit Profitquot2 ProbProfit ProbProfit Z Variance St Dev 02 200 40000 4000 800000 08 50 2500 4000 200000 000 gt E Y 1 I 0 I 1000000 I1000000 I 100 V E2Y VEirzi l oy EY2 E2Y Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter 7 Page 117 EMSE 269 Elements of Problem Solving and Decision Making MaxELQ I Prob Profit ProbProfit 0600 5250 3150 100K 0400 5375 2150 EMV5250K 103900 Dome 06 Low 025 150K EMV10K Km 500K Drill at Site 1 Dry 0850 100K EMV0K Drill at Site 2 EMV5375K Dome 04 Dry 02 Low 0125 High 0025 Low 08 150K 500K 200K 50K 5103 Prob Pro t ProbPro t 0600 10000 600 0250 15000 3751 0150 50000 7501 5251 Prob Pro t ProbPro t 0850 10000 850 0125 15000 1871 0025 50000 1251 537l Prob Pro t ProbPro t 0200 20000 400 0800 5000 4001 001 Instructor Dr J Rene van Dorp Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen Chapter7 Page 118 EMSE 269 Elements of Problem Solving and Decision Making 5103 0 Dominance and making decisions under Uncertainty DETHHKNHDHSTBCINDNHDUUVCE 7 Assume random Variable X Uniformly Distributed on A B Assume random Variable Y Uniformly Distributed on CD X 0 gtPDF w 0 U gt CDF gt39 w Instructor Dr J Rene van Dorp Chapter7 Page 119 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 STOCHASTIC DOMINANCE Assume random Variable X Uniformly Distributed on AB Assume random Variable Y Uniformly Distributed on CD B X Note m Y l I I PrYltz lt PrXlt z T I I I I forallz I I I 0 I I i I A C B D 1 7w LTt 3 Y I 8 X39 I T I I I I O A C B D Instructor Dr J Rene van Dorp Chapter7 Page 120 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 CHOOSE ALTERNATIVE WITH BEST EMV Assume random Variable X Uniformly Distributed on AB Assume random Variable Y Uniformly Distributed on CD 0 gtPDF gt CDF Instructor Dr J Rene van Dorp Chapter7 Page 121 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 MAKING DECISIONS amp RISK LEVEL DETERMH IISTIC DOME IAN CE PRESEQ I I I Chances I of unlucky STOCHASTIC DOME IAN CE PRESENT I outcome I Increases V CHOOSE ALTERNATIVE WITH BEST EDD Instructor Dr J Rene van Dorp Chapter7 Page 122 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Continuous Random Variables 0 Event A with possible outcomes Example 0 A A component s failure o A component fails at time t o S l Total Event or Failure times ranging anywhere from zero to infinity Remember A Random Variable Y Uncertain Quantity is a function from Q R Instructor Dr J Rene van Dorp Chapter7 Page 123 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Define Y failure time of the component Then YltAgtr Often abbreviated to When the number of outcomes of the event A is infinite and uncountable Y is a continuous random variable Instructor Dr J Rene van Dorp Chapter7 Page 124 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Continuous Probability Density function 0 Pryzy 0 for any value of y in the range of Y PI39QEabgtO ifa lt b an ajb falls within the range of Y 0 Probability Density Function fY Z 0 of my 1 i 1 MW gt 0 Instructor Dr J Rene van Dorp Chapter7 Page 125 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 0 dy is a very small interval at y fyydy My lt Y S y dy IfyydyZPryltYltydy y M I I 100 i 100E00 080 800E01 060 l 39 600E01 040 I 400E01 quot6 0quot39quot 200E 01 RM 000 I I 000E00 1 11 21131 41 51 61 71 81 91 I I fltygt PrltYltygt Y Instructor Dr J Rene van Dorp Chapter7 Page 126 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Cumulative Probability Distribution CDF FY y Prat s y l fY udu 1OOEOO 800E01 600E01 400E01 200E01 QOOEOO 11121314151617181 f PrYlt Note CDF is always always a nondecreasing function Instructor Dr J Rene van Dorp Chapter7 Page 127 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen EMSE 269 Elements of Problem Solving and Decision Making 5103 Examples o Exponential Distribution fY A 39 6Xp A 39 o Weibull Distribution fY y a y l eXpa y 0 Beta Distribution F05 05 1 1 l My Faor y y All formulas for Expectation and Variance carry over from discrete case to the continuous case Instructor Dr J Rene van Dorp Chapter7 Page 128 Source Making Hard Decisions An Introduction to Decision Analysis by RT Clemen

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