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# Class Note for EMSE 269 with Professor Dorp at GW (6)

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Makin ar 1 Decisions R T Clemen T Reilly Draft Version 1 Chapter 7 Probability Basics Making Hard Decisions R r Ciemen i REHW Chapter 7 Probability Basics Slide 1 of 62 Le meNmesbv m vanDamandiA Mazzucm conRiGHi 2uu5 millvwwvseasngedul39vdamiv bvevvu Introduction Let A be an event with possible outcomes A1 A 3 n A Flipping a coinquot A1 Heads A2 Tails The total event 9 or sample space of event A is the collection of all possible outcomes of A Q 2 Heads Tails Formally QzAlquunUA UAn U1A Draft Version 1 3 Making Hard Decisions Chapter 7 Probability Basics Slide 2 of 62 R r mm W immensest m cowwem WW Draft Version 1 r clemen r ReillV Making Hard Decisions outcomes in the sample space 9 1 Probabilities must be between 0 and 1 for all possible Le ureNmesby m vanDanandiA Mazzuchi mplwwmseasngedui39vdamm Chapter 7 Probability Basics can be interpreted as the probability ofthe event 0 S PrAl S 1 for all outcomes A that are in S2 Ratio of the area of the oval and the area of the total rectangle by ewu Slide 3 of 62 conRieHr ezuus Probability rules may be derived using VENN DIAGRAMS Probability Calculus S 3 E 395 2 9 5 n a a a Le meNmesbv m vanDamandlA Mazzuchi mpuwwseasgvmuuwamm Chapter 7 Probability Basics bv ewu Slide 4 of 62 copvmw 2uus Draft Version 1 X x A AmAz 2PrltA1qugt PrA1 PrA2 at the same time 2 Probabilities must add up if both events cannot occur Probability Calculus Probability Calculus of these can occur at the same time their Total 3 If A1 An are all the possible outcomes and not two Probability must sum up to 1 Slide 5 of 62 copvmw 2uus bv cvvu e w t s u m 5m X 6 km y me d e w w t r w s c S P 1 m Iu kw 2249 a S V h M H m W cm 21 m m e n r m a a m n 5 A c 3 m H mm A0 H a Mm F gig in Draft Version 1 R r clemen r ReillV Making Hard Decisions the probability of A1 4 The probability of the complement of A1 equals 1 Le ureNmesbv m vanDamandlA Mazzuchi millwwseasngedul39vdamm Chapter 7 Probability Basics PrA1 1 PrA1 by ovvu Slide 6 of 62 conRioHr 2uus mInus Probability Calculus Probability Calculus If two events can occur at the same time the probability of either of them happening or both equals the sum of 5 the probability ofthem Inus their individual probability in both happening at the same time PrA1 PrA2 PrA1 m A2 PrA1 U A2 I uolslal uua Slide 7 of 62 conRicHr 2uus by cvvu Chapter 7 Probability Basics Le ureNmesbv m vanDamandlA Mazzuchi r Clemenil ReiiW Making Hard Decisions mp WNWseas ng eduldavpiv Probability Calculus 6 Conditional probability Dow Jones Up New Total Event based on the condition that we know that the Dow Jones went up Stock Price Up Draft Version 1 39 Making Hard Decisions Chapter 7 Probability Basics Slide 8 of 62 R r clemen r Reilly Le uveNmesbv m vanDamandlA Mazzuchi conRiGHrezuus NipWmvseasngedul39vdamn bvevvu Probability Calculus Conditional Probability PrStock T Dow T PrD0w T Intuition Ifl know that the market as a whole will go up the chances of the stock of an individual company going up will increase PrStock Tl Dow T PrA n B PrA B MB lnformally Conditioning on an event coincides with reducing the total event to the conditioning event Draft Version 1 Making Hard Decisions Chapter 7 Probability Basics Slide 9 of 62 r C emen l my Le uveNmesbv m Mazzuchi covwewmuua Mullvwwvse bvcvvu Probability Calculus Conditional Probability Example The probability of drawing an ace of spades in a deck of 52 cards equals 152 However if I tell you that I have an ace in my hands the probability of it being the ace of spades equals PrAce Space 152 1 PrAce 4 52 4 PrSpades Ace 2 Note also that PrA n B PrB A PM Draft Version 1 Making Hard Decisions Chapter 7 Probability Basics Slide 10 of 62 r Clemen l Reilly Le uveNmeshv m Mazzuchi covwewmuua Niplvwwvse bvcvvu Probability Calculus 7 Multiplicative Rule Calculating the probability of two events happening at the same time PrAl m B PrB A gtlt PrA PrA B gtlt PrB 8 Independence between two events Informally two events are independent if information about one does not provide you any information about the other and vice versa Consider Event A with possible outcomes A1An g Event B with possible outcomes BlBm Making Hard Decisions Chapter 7 Probability Basics Slide 11 of 62 r clemen r Remy Le meNmesbv J copvmewezuus Niplvwwv bvevvu Probability Calculus Independence Example A is the event of flipping a coin and B is the event of throwing a dice If you know the outcome of ipping the coin you do not learn anything about the outcome ofthrowing the dice regardless of the outcome of ipping the coin Hence these two events are independent Formal definition of independence between event A and event B PrAr lBj PrA For all possible combinations and B Draft Version 1 E Making Hard Decisions Chapter 7 Probability Basics Slide 12 of 62 a r cam my Le uveNmesbv J conRicHi 2uus MPIvwwv bvcvvu Probability Calculus Independence Equivalent definitions of independence between A event and event B 1 PrBJ Ai PrBJ For all possible combinations and B 2 PrAl n Bf PrAlgtlt PrBj For all possible combinations and B Independencedependence in influence diagrams No arrow between two chance nodes implies independence between the uncertain events An 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 likelihood of outcomes of B Draft Version 1 39 Making Hard Decisions Chapter 7 Probability Basics Slide 13 of 62 R r Clemen l Reilly Le meNmesbv m vanDamandlA Mazzuchl conRchrezuus mPlwwwseasngedul39vdamlvl bvcvvu Probability Calculus Conditional Independence Example The performance of a person on any IQ test is uncertain and may range anywhere from 0 to 100 However if you to know that the person in question is highly intelligent it is expected hisher score will be high eg ranging anywhere from 90 to 100 On the other hand the person s IQ does not explain this remaining uncertainty and it may be considered measurement error affected by other conditions For example having a good night sleep during the previous night On any two IQ tests these measurement errors may be reasonably modeled as independent if we know the IQ ofthe person 2 t gt it Lquot o Making Hard Decisions Chapter 7 Probability Basics Slide 14 of 62 r Harmful Reilly Le meNmesbv J conRIcHrezuus NIPlvwwv bvcvvu Draft Version 1 Probability Calculus Conditional Independence Event A with possible outcomes A1A Event B with possible outcomes BlBm Event C with possible outcomes C1Cp Formal definition Event A and event B are conditionally independent given event C if and only if PrAl B Ck PrAl Ck For all possible combinations B and Ck lnformally Ifl already know C any information or knowledge about B does not tell me anything more about A Making Hard Decisions Chapter 7 Probability Basics Le uveNmesbv J Niplvwwv Slide 15 of 62 copvmew zuus hvGWU r clemen r ReilW Probability Calculus Conditional Independence Equivalent definitions Event A and event B are conditionally independent given event C if and only if 1 PrBj AiCkPrBj Ck For all possible combinations B and Ck 2 PrAl n3 Ck PrAl CkgtltPrBj Ck For all possible combinations A B and Ck Draft Version 1 Making Hard Decisions Chapter 7 Probability Basics Slide 16 of 62 v Clemen l Reilly Le uveNmeshv m Mazzucm conRicHtezuus Niplvwwvse bvcvvu Probability Calculus Conditional Independence Equivalent definitions Event A and event B are conditionally independent given event C if and only if 1 PrBj AiCkPrBj Ck For all possible combinations B and Ck 2 PrAl n3 Ck PrAl CkgtltPrBj Ck For all possible combinations A B and Ck Draft Version 1 Making Hard Decisions Chapter 7 Probability Basics Slide 17 of 62 v Clemen l Reilly Le uveNmeshv m Mazzucm conRicHtezuus Niplvwwvse bvcvvu Probability Calculus Conditional Independence Conditional independence in influence diagrams C Draft Version 1 39 Making Hard Decisions Chapter 7 Pr Lena R r Clem r Reilly W m ity Basics Slide 18 of 62 copvmew e was vaWU R r Clemen i My Making Hard Decisions Le uveNmesbv m vanDaipandiA Mazzuchi mpiwwseasngedui39vdamlvi Chapter 7 Probability Basics by cvvu Slide 19 of 62 conRieHi ezuus Draft Version 1 PrA PrA n 131 PrAnBzPrAnB3 ltgt PrA PrA BlPrBlPrA BzPrBzPrA B3PrB3 LetBl B3 be mutually exclusive collective y exhaustive Probability Calculus Law of Total Probabi ity Probability Calculus Law of Total Probability Example X System fails SYSTEM X X failure X No Failure A Component A fails B Component B fails B C Component C fails Draft Version 1 Assume that components A B and C operate independently 39 Making Hard Decisions Chapter 7 Probability Basics Slide 20 of 62 R r ciemen r Reilly Le uveNmesbv m vanDamandlA Menu2m conRiGHrezuus mplwmvseasngeduldavpiv bvevvu Probability Calculus Law of Total Probability Task Write the probability of failure PrX as a function of the component failure probabilities PrA PrB and PrC 1PrX PrX APrA PrX ZPrZ 1PrAPrX ZPrZ 2PrX Z PrX BZPrB Z PrX ZPr Z PrX BAPrB0PrB PrX BZPIB Substitute result 2 into 3 Draft Version 1 3PrX PrA PrX B ZPrBPrA7 Making Hard Decisions Chapter 7 Probability Basics Slide 21 of 62 r Clemen f Reilly Le meNmesbv m van de Menu2m copvmewezuus Niplvwwvseasg bvevvu Probability Calculus Law of Total Probability Intermediate conclusion Hence we need to further develop PrX B Z 4PrX BZ PrX CBZPrC BZ PrX BA7Pr 32 1PrC 0 Pr PrC Substitute result 4 into 3 5PrX PrA PrCPrBPrA7 6PrZ1 PIA Substitute result 6 into 5 Draft Version 1 7PrX PrAPrCPrB PrCPrBPrA Making Hard Decisions Chapter 7 Probability Basics Slide 22 of 62 r Clemen l Reilly Le meNmesbv m van dlA Mazzucm copvmewezuus Niplvwwvseasg bvcvvu Probability Calculus Law of Total Probability Example Oil Wildcatter Problem M ax P ro t Dry 0 100K Drill at Site 1 150K 500K 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 Draft Version 1 39 Making Hard Decisions Chapter 7 Probability Basics Slide 23 of 62 R r Hemeer Remy Le ureNmesbv m vanDarpande Menu2hr conchHrezuus mPlwwseasngedudarprr bvcvvu Draft Version 1 Probability Calculus Law of Total Probability PriDome PrlNo Dome 0600 0400 Outcome PrloutcomelDome Outcome PrloutcomelNo Dome Dry 0600 Dry 0050 Low 0250 Low 0125 High 0150 High 0025 Making Hard Decisions Chapter 7 Probability Basics Slide 24 of 62 R r Harmful REHW Le meNmesbv m vanDamandlA Mazzucm mp lvwwvseas ng 200mm copvmew 2005 hvGWU Probability Calculus Law of Total Probability PrDry PrDry DomePrDome PrDry No DomePrNo Dome 0600 0600 0850 0400 0700 PrLow PrLow Dome PrDome PrLow No DomePrNo Dome 0250 0600 0125 0400 0200 PrHigh PrHigh DomePrDome PrHigh No DomePrNo Dome 015006000025 0400 0100 Draft Version 1 Making Hard Decisions Chapter 7 Probability Basics Slide 25 of 62 r memem Reilly Le meNmesbv m van dlA Mazzucm copvmwmuus Niplvwwvseasg bvevvu Probability Calculus Law of Total Probability Dry 0600 100K Dome 0600 Low 0250 150K 500K 100K 150K Low 0250 0600 0125 0400 020 150K 39 h 0150 0600 0025 0400010 500K Draft Version 1 Informally when we apply LOTP we are collapsing a probability tree Making Hard Decisions Chapter 7 Probability Basics Slide 26 of 62 R r cwemem REHW Le meNmesbv m vanDamandlA Menu2m copvmewezuus Niplvwwvseasngedul39vdamn bvevvu R r Clemen i ReilW Making Hard Decisions Le uveNmesbv m vanDaipandiA Mazzuchi mpiwwseasngedui39vdamlvi Chapter 7 Probability Basics by ovvu Slide 27 of 62 conRieHi 2uus Draft Version 1 1 From the multi pli cative rule it follows that PrAnBj PrBj APrA PrA BjPrBj Let 31 B3be mutually exclusive collectively exhaustive Probabi ity Calculus Bayes Theorem Probability Calculus Bayes Theorem 2 Dividing the LHS and RHS of 1 by PrA yields PrA B PrBj Pr B A PrA 3 We may rewrite PrA using the Law of Total Probability yielding PrA PrA I Bl PrBl PrA le PrBz PrA I B3 PrB3 4 Substituting the result of 3 into 2 gives perhaps the most well known theorem in probability theory Draft Version 1 Bayes Theorem Making Hard Decisions Chapter 7 Probability Basics Slide 28 of 62 r Ciemen Remy rammest m Mazzucm conRieHrezuus Nipivwwvse hvevvu Probability Calculus Bayes Theorem PrA BPrB Pme A PrA Bl PrBl PrA BzPrBz PrA B3PrB3 Thomas Bayes 1702 to 1761 Bayes theory of probability was pub ished in 1764 His conclusions were accepted by Lap ace in 1781 rediscovered by Condorcet and remained unc1aenged until Boole questioned them Since then Bayes39 techniques have been sub39ect to controversy Draft Version 1 Source httpwww gapdcsstandacukNhistoryMathematiciansBayeshtml Making Hard Decisions Chapter 7 Probability Basics Slide 29 of 62 R r Clemen l Reilly Le uveNmeshv A Mazzuchi conRicHrezuus hvGWU Probability Calculus Bayes Theorem Oil Wildcatter Problem Example Continued 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 1 From the rule for conditional probability it follows that PrHighiDomePrD0me PrD0me High W 2 From the LOTP it follows that PrHigh PrHigh Dome PrD0me PrHighi N0 Dome PrNo Dome Draft Version 1 3 Making Hard Decisions Chapter 7 Probability Basics Slide 30 of 62 a r Ciemen f Remy Le uveNmesbv J conRicHrezuus Niplvwwv hvevvu Draft Version 1 Probability Calculus Bayes Theorem Oil Wildcatter Problem Example Continued 3 Substitution of 2 in 1 yields PrDomel High Pr IIigthame Pr Dame PrfIigthamePr D0mePrfIigth0 D0mePr N0 Dome 01500600 015006000025004OO 090 PrDome The Prior Probability PrDomelData The Posterior Probability Data The well is a high producesquot Making Hard Decisions R r Clemen l REHW Chapter 7 Probability Basics Lennie Nmes W A Mamcm Slide 31 of 62 copvmew e zuus vaWU Probability Calculus Bayes Theorem Draft Version 1 Oil Wildcatter Problem Example Continued Notice that PrDryPrLow and PrHigh have been inserted in the treeThese were calculated using LOTP Notice that PrDomeHigh has been inserted as well This one was calculated using Bayes Theorem We need to fill out the Remainder of the question Marks Dame a BBB Nu Dame El ADD Dry a BBB VWDEIK Law a 25m 15m ADM 15M SEEM BAYES TH EOR EM Dry u 7 Hrgh u m ADM ADM 15M 15M SEEM SEEM Making Hard Decisions R r Clemenrl Reer Chapter 7 Probability Basics rammest m vanDamandiA Mazzuchr mpuwmeasgmwmmm Slide 32 of 62 conchHr zuus hvcvvu Probability Calculus Bayes Theorem Oil Wildcatter Problem Example Continued When we reverse the order of the chance nodes in a decision tree we need to apply Bayes Theorem PrDome PrNo Dome 0 600 0 400 x PrXDome PrXNo Dome PrX n Dome PrX n No Dome PrX PrDomeX PrNo DomelX Check Dry 0 600 0 850 0 360 0 340 0 700 0 514 0 486 1000 Low 0 250 0125 0150 0 050 0 200 0 750 0 250 1000 High 0150 0 025 0 090 0 010 0100 0 900 0100 1000 Check 1000 1000 0 600 0 400 1000 E E E Next we allocate the probabilities from the table at 5 their appropriate locations in the tree Chapter 7 Probability Basics Slide 33 of 62 3 Making Hard Decisions R r memen l Rexllv Le uveNmesbv m vanDamandlA Mazzucm mp lvwwvseas ng eduldamlU copvmew e zuus hvcvvu Probability Calculus Bayes Theorem Oil Wildcatter Problem Example Continued AFTER BAYES THEOREM Dome 0514 100K No Dome 0486 100K Dome 0750 150K No Dome 0250 150K Dome 0900 g 500K E High 010 5 No Dome 0100 500K 39 Making Hard Decisions Chapter 7 Pr ity Basics Slide 34 of 62 copvmew 0 2005 hvGWU Probability Calculus Bayes Theorem The Game Show Example Suppose we have a game show host and you There are three doors and one ofthem 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 Door1 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 Draft Version 1 WHAT SHOULD YOU DO 3 Making Hard Decisions Chapter 7 Probability Basics Slide 35 of 62 Mam R r ciemenr Remy Le uveNmeshv m conRicHrezuus mPlvwwvse by cvvu Probability Calculus Bayes Theorem The Game Show Example 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 Define Di Prize is behind door i i1 Hi Host shows Door i containing no prize after you selected Door 1 i1 Draft Version 1 1 1 It seems reasonable to set prior probabilities PrD 3 Making Hard Decisions Chapter 7 Probability Basics Slide 36 of 62 r Clemen l my Le uveNmeshv J cowewmuua Niplvwwv bvovvu Draft Version 1 Probability Calculus Bayes Theorem 2 Apply LOTP to calculate PrH3 PrH3 ZPrH3 DiPrDi i1 llll0ll 2 3 3 3 2 3 Calculate PrD1H3 11 PrD1H3W 2 3 1 PrH3 l 3 4 Apply the complement rule 2 1 2 PrD2lH31 PrDIH31 80 YES you should SWITCH since you would increase your chances of winning Making Hard Decisions Chapter 7 Probability Basics Le uveNmesbv J Niplvwwv Slide 37 of 62 copvmew e was vaWU r Clemen l ReilW Probability Calculus Uncertain Quantities Example When a student attempts to log on to a computer timesharing system either all ports are busy B in which case the student will fail to obtain access or else there is at least one port free F in which case the student will be successful in accessing the system Total Event S2 RF Definition For a given total event 2 a random variable rv is any rule that associates a number with each 2 outcome in In mathematical language a random variable is a function whose domain is the total event and whose range is the real numbers Draft Version 1 3 Making Hard Decisions Chapter 7 Probability Basics Slide 38 of 62 a i cieneni Remy Le uveNmeshv JR conRiGHiezuus hvGWU 3g 5 390 9 m 0 N 25 2 8 m W83 1 39z 9 C D 0 a 3 C m a n 3 v 0 39o L 0 II S m m 5 I53 3 5 a V g 4 a t i N C 39O m 2 D 1 a g 2 i II 5 E A 33 E U D 55 A 4 M D 3 L14 g g m g V a b 3g 0 Q W 75 MS 2 N 5 a n 5 a a V c O 9 390 8 5 8 g gt 53 E 3 2 ivv g Lu w o 39 g 1 o N o 3 g 5 a a 390 O 393 Q a l 8 o D C gt 5 quot39 E E E n m 3 61 X x U E LIJ a m g luolSJSAJJEJG Probability Calculus Uncertain Quantities Define a rv X as follows X The number of batteries examined before the experiment terminates Then XG 1 XBG 2 XBBG 3 etc The argument ofthe random variable function is typically omitted Hence one writes PrX 2 PrSecond Battery Works Note that the above statement only has meaning with the above definition ofthe random variable It is good practice to always include the definition of a random variable in words Draft Version 1 Making Hard Decisions Chapter 7 Probability Basics Slide 40 of 62 r Clemen Reilly Le uveNmeshv J cowewmuua Niplvwwv bvcvvu Probability Calculus Uncertain Quantities The nature of random variables can be discrete and continuous Definition A discrete random variable is an rv whose possible values either constitute a nite set or else can be listed in an in nite sequence in which there is a rst element a second element and so on Think of the previous batter example A random variable is continuous if its set of possible values consists of an entire interval on the number line For example the failure time of a component Draft Version 1 39 Making Hard Decisions Chapter 7 Pr Lena R r Clem r Reilly W m ity Basics Slide 41 of 62 conRchr mus hvGWU Discrete Probability Distributions The nature of random variables can be discrete and continuous Definition A discrete random variable is an rv whose possible values either constitute a nite set or else can be listed in an in nite sequence in which there is a rst element a second element and so on Think of the previous batter example A random variable is continuous if its set of possible values consists of an entire interval on the number line For example the failure time of a component Draft Version 1 39 Making Hard Decisions Chapter 7 Probability Basics Slide 42 of 62 R r clemem Reilly Le meNmesbv m vanDamandlA Mazzuchl conRchrezuus Niplvwwvseasngedul39vdamn bvcvvu Discrete Probability Distributions Example Y Number of Raisins in an Oatmeal Cookie Assume possible outcomes Y1 Definition The probability mass function PMF on is the collection of probabilities such that PrYipi Y 1 2 3 4 5 PrYi 01 015 03 035 01 Note that PrY 1 PrY 2 PrY 3 PrY 4 PrY 5 iPrYi1 Draft Version 1 Making Hard Decisions Chapter 7 Probability Basics Slide 43 of 62 r cwemem Reilly Le meNmesbv m copvmewezuus vaWU Discrete Probability Distributions 2 t gt ti Lquot o Graphical Depictions of PMF s n1 n4 n5 n4 77777777777777777777777777777777777777777 77 as W as Pi DZ P M n1 A Histogram An Line Graph Definition The cumulative distribution function CDF on at y is the sum of the probabilities such that Ys y r ciemen r REHW Making Hard Decisions FyPrY s y Z PrY i iiSy Slide 44 of 62 copvmew zuus vaWU Chapter 7 Probability Basics Le uveNmesbv J R Mazzucm Niplvwwvse Discrete Probability Distributions Graphical Depictions of CDF 1 08 06 Fy 04 02 In Decision Analysis a CDF is referred to as a CUMMULATIVE RISK PROFILE and a PMF is referred to as a RISK PROFILE Draft Version 1 39 Making Hard Decisions Chapter 7 Probability Basics Slide 45 of 62 R r cwemem Reilly Le meNmesbv m vanDamandlA Mazzucm copvmewezuus mpwmvseasngeduldavmv bvcvvu 2 t gt ti Lquot o Probability Calculus Expected Value 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 expected value of Y Interpretation If you were able to observe many outcomes on the calculated average of all the outcomes would be close to EY If 2 gY n n EZ ZgltyigtxPrltY y 2 gltyigtxpi Making Hard Decisions r clemem RelllY Slide 46 of 62 conRchr zuus hvGWU Chapter 7 Probability Basics Le uveNmeshv J R Mazzuchl Niplvwwvse Probability Calculus Expected Value Draft Version 1 If ZaYb ab constants Y a rv EZ aEY b If ZaXbY ab const XY a rv EZ aEXbEY Oatmeal Cookie Example 1Raisin 010 Raisins RaisinsPrY Raisins 1 1 0 100 10 320 2 2 2015030 3 Raisins 030 3 3030090 4 4 0 351 40 4 5 5 0 1 00 50 5 Raisins 010 5 320 On average an oatmeal cookie has 32 Raisins Making Hard Decisions R r cwemem REHW Chapter 7 Probability Basics Le uveNmeshv m vanDavpandiA Mazzucm Nipivwwvseasngedui39vdamm Slide 47 of 62 copvmw 2uus vaWU Variance and Standard Deviation 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 ie the MAX VALUE on minus the MIN VALUE on However this completely ignores that some values on may be more likely than others SUGGESTION Calculate the BEST GUESSquot for the DISTANCE from the MEAN The standard deviation can be thought of such a guess The standard deviation on is the square root of the variance on Draft Version 1 3 Making Hard Decisions Chapter 7 Probability Basics Slide 48 of 62 man a i Clememi Reilly Le uveNmesbv JR conRiGHiezuus Mullvwwvse by cvvu Variance and Standard Deviation Variance VarY 0 EY EY2 EY2 2YEY E2Y EY2 2EYEY E2Y EY2 E2Y Standard Deviation ay 40 JEY2 E2Y Draft Version 1 7quot Making Hard Decisions Chapter 7 Probability Basics Slide 49 of 62 Ciemen i Remy Le uve zzzzz m conRieHrezuus bv ewu Variance and Standard Deviation If ZaYb ab constants Y a rv Var Z 2 anar Y If ZaXbY ab const XY indepent rv s VarZ anarX b2VarY Max Pro t Example 024 20 3575 A g W Note that 35 9 50 EA EB F 025 5575 399 PrPro t s 0 l A o g B 035 0 5 PrPro t s 0 l B 06 040 95 Making Hard Decisions Chapter 7 Probability Basics Slide 50 of 62 v c5555 5550 55555051555 1 conRleHrezuus 51 qu hvevvu Variance and Standard Deviation Draft Version 1 R r clemem ReilW Le meNmesbv m vanDamandYA Mazzucm mtplvwvwseasngeduldamlv Alternative A Prob Profit Profitquot2 ProbProfit ProbProfitquot2 Variance St Dev 024 20 400 480 9600 047 35 1225 1645 57575 029 50 2500 1450 72500 am 3575 12780625 139675 11869 1089438 E IY EIY ElY JrE m 71 Alternative B Prob Profit Profitquot2 ProbProfit ProbProfitquot2 Variance St Dev 025 9 81 225 2025 035 0 0 000 000 04 95 9025 3800 361000 3575 12780625 363025 235219 4849936 39 Making Hard Decisions Chapter 7 Probability Basics Slide 51 of 62 conRloHr mus hvGWU Expected Value Variance and Standard Deviation Max Pro t Dry 07 100K Drill at Site 1 150K High 01 500K Example D 0 2 ry OII Wildcatter 200K Problem Continued Drill at Site 2 50K Drill at Site 1 Prob Profit Profitquot2 ProbProfit ProbProfitquot2 Variance St Dev 07 100 10000 7000 700000 02 150 22500 3000 450000 3 0 1 500 250000 5000 2500000 E 1ooo gtEY E I 100 1 3650000 3640000 1907878 5 l EZY EUZ 1 l UY EYZ7EZY 3 Making Hard Decisions Chapter 7 Probability Basics Slide 52 of 62 R r meme Rem Lemwa m mm Mamm conRicHrezuus Niplwwmseasngedul39vdamyv hvcvvu Expected Value Variance and Standard Deviation Draft Version 1 Max Pro t Dry07 100K Drill at Site 1 150K H39 h 01 lg 500K Example D 0 2 ry OII Wildcatter 200K Problem Continued Drill atSite 2 50K Drill at Site 2 Prob Profit Profitquot2 ProbProfit ProbProfitquot2Variance St Dev 02 200 40000 4000 800000 08 50 2500 4000 200000 000 gtEY I O 1 1000000 1000000 I 100 Ezm EYZ 1 ay EYZEZY Making Hard Decisions Chapter 7 Probability Basics Slide 53 of 62 r cwemem ReHW Le meNmesbv m vanDamandYA Menu2m copvmew e was mp lvwwvseas ng eduldavmn bv ewu Expected Value Variance and Standard Deviation Max Pm t Div U 6D 40014 Prob Pro t Prob Pro t EMV52 5014 D U6 0600 10000 6000 W 15014 0250 15000 3750 0150 50000 7500 EMV1EIK H gh D 15 SEEM 5250 DHH at stew r my D 55539 ADEIK EMV53 75K Prob Pro t Prob Pro t ND DUNE D A W U 125 0850 10000 8500 X 15 0125 15000 1875 EMV Prob Pro t Prob Pro t 0025 50000 1250 cm 0500 5250 3150 High U U25 5W W 0400 5375 2150 I 1000 Dry 0 2 7200K Prob Pro t Prob Pro t 0200 20000 4000 0800 5000 4000 55 000 Drm at Ste 2 Draft Version 1 Expected Values can be calculated by folding back the tree Making Hard Decisions Chapter 7 Probability Basics Slide 54 0f 62 R r cwemem Remy Le uveNmesbv m vanDamandYA Mazzucm copvmwmuus mpwmvseasngeduldavmv bvcvvu Draft Version 1 Continuous Probability Distributions Let X The failure time of a component Definition Let X be a continuous rv Then a probability density function pdf of X is a function fx such that for any two numbers a and b with a lt b For fx to be a legitamate b Pr XE ab x dx D 1 pdf we must have fx fx 2 x for all possible values x fxzlx 1 Making Hard Decisions Chapter 7 Probability Basics Le uveNmeshv J hmme Slide 55 of 62 copvmew e zuus vaWU r Clemen l REHW 2 t gt ti Lquot o Continuous Probability Distributions Cumulative distribution function fltugt F x PrX S x I f udu The pth quantile xp fltxgt Fxp PrX s xp p I gt Making Hard Decisions Slide 56 of 62 R r Clemen l Reillv copvmew zuus vaWU Chapter 7 Probability Basics Le uveNmesbv J Mazzucm Niplvwwv v Continuous Probability Distributions Let X The failure time of a component fltXgt Pra SX Sb x Fx PrX s a Draft Version 1 Conclusion Pra s X s b PrX s b PrX s a Fb Fa Making Hard Decisions Chapter 7 Probability Basics Slide 57 of 62 h R r Ciemen i Remy Le meNmesbv J M zzzz m copvmwmuus NipWWW y bv ewu Draft Version 1 Continuous Probability Distributions Examples theoretical density functions 1 ix H2 Normal fx 05 6XP 2 02 axe 1R Exponential fx xi eXp 1 x x gt 0 Fa8 171 rl Beta fx rar x 1 x xe 01 More in Chapter 9 Be Expected Value Ele J xfxtl Variance VarX i x EX2fxtlx Formulas carry over from the discrete to the continuous case Making Hard Decisions Chapter 7 Probability Basics Le uveNmeshv J Niplvwwv Slide 58 of 62 copvmew e was vaWU r Ciememl REHW Dominance Revisited DETERlVHNISTIC DOlVHNANCE Assume random Variable X Uniformly Distributed on AB Assume random Variable Y Uniformly Distributed on CD a X i Y O lt I 39 I I I 0 I i I l A B C D Ln 1 quotquotquotquotquotquotquotquotquotquotquotquot quot7 quotquotquot 7 Q X 39 Y x E 0 x T x Z 0 I I Ir I g A B C 3 Making Hard Decisions Chapter 7 Probability Basics Slide 59 of 62 Dominance Revisited STOCHASTIC DOMINANCE Assume random Variable X Uniformly Distributed on AB Assume random Variable Y Uniformly Distributed on CD a X 39 Note A Y l I I PrYltz lt PrXlt z I I I T I I 39 I for all z I l I I I I I 0 I I I I A C B D Lu Q a o 2 T if a 6 3 Making Hard Decisions Chapter 7 Probability Basics Slide 60 of 62 a r mm W rammest m WWW Mam conRieHrezuus marrwsmmarwi WW Dominance Revisited CHOOSE ALTERNATIVE WITH BEST EMV Assume random Variable X Uniforrnly Distributed on AB Assume random Variable Y Uniforrnly Distributed on CD EX EY O gtPDF gt CDF Draft Version 1 3 Making Hard Decisions Chapter 7 Probability Basics Slide 61 of 62 R r am Mr Le meNmesbv m vanDamandYA Mazzuchi conRiGHrezuus mPlvwwvseasngeduldamiv hvevvu Making Decisions under Uncertainty Deterministic Dominance Present Chances of an unlucky Stochastic Dominance Present g outcome 39 increase F Making Decisions based on EMV gt 5 a 39 Making Hard Decisions Chapter7 Probability Basics Slide 62 of 62 Niplvwwwseasngedui39vdamlvl bvcvvu

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