Taking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling


 Buck Blair
 3 years ago
 Views:
Transcription
1 Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria {maheswar, tambe, bowrig, jppearce, DCOP Formulatios for DiMES: Proofs of Propositios TSA Time Slots as ariables): This method reflects a atural first step whe cosiderig schedulig issues. Let us defie a DCOP where a variable x t) represets the th resource s tth time slot. Thus, we have N T variables. Each variable ca a tae o a value of the idex of a evet for which it is a required resource, or the value 0 to idicate that o evet will be assiged for that particular time slot: x t) {0} { {1,..., K} : R A }. It is atural to distribute the variables i a maer such that {x 1),..., x T)} belog to a aget represetig the schedule of the th resource. Propositio 1 The DCOP formulatio with time slots as variables, where the costrait betwee variables x 1 t 1 ) ad x 2 t 2 ) whe x 1 t 1 ) = ad x 2 t 2 ) = 2 taes o the utility f 1, t 1, ; 2, t 2, 2 ) = ) ) ) I {1 2 } 1 I{t1 t 2 } 1 I{P1 P 2 A {1,...,K}} fiter, 2 ) + I {1 = 2 } 1 I{t1 =t 2 } fitra 1 ; t 1, ; t 2, 2 ) where the iteraget costrait utilities are f iter, 2 ) = MI {1 2 }I {A A 2 } ad itraaget costraits fully coected amog the time slots of a sigle aget) for t 1 < t 2 w.l.o.g. are f itra ; t 1, ; t 2, 2 ) = M 0, t 2 t 1 < L, 2 M 0, t 2 t 1 L, 2 = g; t 1, ; t 2, 2 ) otherwise where g; t 1, ; t 2, 2 ) = 1 [ 0 t 1 ) ) I {1 0} + 2 i 0 T 1 t 2) ) ] I {2 0} with M > NT max where N is the umber of participats, T is the umber of time slots ad max = max,, yields the optimal solutio to the Distributed MultiEvet Schedulig DiMES) problem. ) Proof. The fuctio f ca be aalyzed as follows. The first term i the factor I {1 2 } 1 I{t1 t 2 } idetifies a iteraget costrait ad the secod term reflects that iteraget costraits exist oly across idetical time slots. The factor 1 I {t1 t 2 }) implies that a iteraget li exists betwee x1 t) ad x 2 t), oly if there exists a evet that requires both P 1 ad P 2 as attedees. The factor I {1 = 2 } idetifies a itraaget costrait ad subsequetly ) 1 I {t1 =t 2 } expresses that there is o costrait from a variable to itself. The fuctio f iter ) idicates that a pealty of M is assessed if the agets assig differet evets for the same time 2 ) ad the evets force a participat to be at two evets at the same time A A 2 ). 1 1 For otatioal cosistecy i the situatio where a agets decides ot to assig a evet to a particular time slot, we defie A 0 :=. 1
2 The fuctio f itra ) assesses a pealty to esure that at least L slots must be assiged cotiguously whe evet E is scheduled 0, t 2 t 1 < L, 2 ). Also, a pealty uder the secod coditio 0, t 2 t 1 L, 2 = ) is assessed to esure that o more tha L slots are assiged ad also to esure that the same evet is ot scheduled twice. If oe of these coditios are met, the utility o the costrait is the differece i the values for attedig the evets ad the value of eepig the time slots uassiged divided by T 1 the umber of outgoig itraaget lis). The sum of all costrait utilities excludig the pealties) is N T 1 =1 s=1 t=s+1 T g; s, x i s); t, x t)) N T 1 T =1 s=1 t=s+1 2 max N T 1 = =1 N T 1 T =1 s=1 t=s+1 1 [ x i s) T 1 ] + x it) TT 1) 2 max 2 T 1 = NT max = M Thus, the total utility if a pealty is icurred is opositive. Cosequetly, it would be better to assig a value of 0 to all variables ad obtai a global utility of zero tha cosider a solutio with a pealty. This implies that ay optimal solutios to the DCOP will ot icur ay pealties. The absece of iteraget pealties imply, for ay, {t : x 1 t) = E } = {t : x 2 t) = E }, 1, 2 such that P 1, P 2 A. This expresses that if a evet is scheduled all participats will have assiged the idetical set of time slots to that evet. The absece of itraaget pealties implies that if P A, {t : x t) = E } = {t0,..., t 0 + L 1} where t0 = mi{t : x t) = E }. This expresses that a evet will be scheduled i cotiguous slots totalig the exact legth of the evet. If we defie S E ) := {t : x t) = E } where P A, the because two evets caot be assiged to the same time slot, we have S E ) S E 2 ) =, 2 {1,..., K}, 2, A A 2. 1) Summig over the lis, we have the followig global utility: N T 1 T =1 s=1 t=s+1 1 [ x s) 0 s) ) I {x s) 0} + x t) 0 t) ) ] I {x t) 0} T 1 Because the itraaget variables are fully coected, each variable has T 1 outgoig lis. Rewritig the previous expressio as a sum over time slots, we have the global utility as: N T x t) 0 t) ) N K I {x t) 0} = I {i A } 0 t) ) K = R t) ). 2) =1 t=1 =1 =1 t S E ) =1 A t S E ) The first equality i 2) is obtaied by partitioig the time domai for each participat ito scheduled evets ad the secod equality by switchig the order of the first two summatios. The solutio to the DCOP will determie a schedule that maximizes the fial expressio i 2) which, whe coupled with the coditios i 1) implied by the absece of pealties, is idetical to the multievet schedulig problem MESP). EA Evets as ariables): We ote that the graph structure of TSA grows as the time rage cosidered icreases or the size of the time quatizatio iterval decreases, leadig to a deser graph. A alterate approach is to cosider the evets as the decisio variables. Let us defie a DCOP where the variable x represets the startig time for evet E. Each of the K variables ca tae o a value from the time slot rage that is sufficietly early to allow for the required legth of the evet or 0 which idicates that a evet is ot scheduled: x {0, 1,..., T L + 1}, = 1,..., K. If a variable x taes o a value t 0, the it is assumed that for all required resources of that evet A ), the time slots {t,..., t + L 1} must be assiged to the evet E. It would be logical to assig each variable/evet to the aget of oe of the required resources for the evet. Propositio 2 The DCOP formulatio with evets as variables where a costrait betwee variables x ad x 2 whe x = t 1 ad x 2 = t 2 exists if the two evets have a commo required attedee A A 2 for 2 ), ad 2
3 cosequetly taes o the utility f, t 1 ; 2, t 2 ) = M t 1 0, t 2 0, t 1 t 2 t 1 + L M t 1 0, t 2 0, t 2 t 1 t 2 + L 2 g, t 1 ; 2, t 2 ) otherwise. where g, t 1 ; 2, t 2 ) = β I {t1 0} L 0 t 1 + l 1) ) + β 2 I {t2 0} A l=1 L 2 A 2 l=1 2 0 t 2 + l 1) ) for β i = 1/ K=1 I {A A i } 1 ) multiplicative iverse of the umber of outgoig lis) with M > NT max where N is the umber of participats, T is the umber of time slots ad max = max,, yields the optimal solutio to the Distributed MultiEvet Schedulig DiMES) problem. Proof. The first two coditios of the fuctio f state that a pealty of M is assessed if the variable assigmets cause a schedulig coflict. If o pealties are imposed, the utility gai for schedulig the evet the differece betwee the rewards for all the attedees a the values for eepig the utilized times uassiged) is distributed uiformly over the outgoig lis from the variable. Let us assume that a pealty is icurred because of a schedulig coflict betwee the variables x = t 1 0 ad x 2 = t 2 0. By settig x = 0 decidig ot to schedule evet E ), the global utility will chage by at least β 2 L 2 2 A 2 l=1 0 t 2 + l 1) ) K=1 I {A A i } 2 L K=1 I {A A i } 1 A l=1 0 t 1 + l 1) ) M where the term i the first set of bracets represets the value of the li whe x = 0 ad the term i the secod set of bracets represets the value of the lie whe x = t 1 0. We ca see that the utility chage is greater tha L M > M NT max > 0. A l=1 Thus, the optimal solutio to the DCOP caot have ay pealties o the lis as we ca always mae a utility gai by ot schedulig a evet that leads to a pealty. Let us defie S E ) = {t,..., t + L 1} if t 0 ad S E ) = if t = 0. Because t + L 1 T which S E ) T. Furthermore, we ow that the optimal solutio will have o coflicts, i.e. S E ) S E 2 ) =, 2 {1,..., K}, 2, A A 2. 3) Sice we have o pealties, the uiformly distributed gais from schedulig ca be aggregated. The global utility will be the sum of the rewards for the scheduled evets mius the values of the utilized times: K L 0 t + l 1) ) = K 0 t) ). =1 A l=1 =1 A t S E ) The DCOP solutio maximizes the previous expressio. This alog with coditio 3), is idetical to the MESP problem. PEA Private Evets as ariables): We ote that i EA, if a aget is to mae a decisio for a evet as a variable, it must be edowed with both the authority to mae assigmets for multiple resources as well as have valuatio iformatio for all required resources. There are settigs where resources, though part of a team, are uwillig or uable to cede this authority or iformatio. To address this, we cosider a modificatio of EA that protects these iterests. Let us defie a set of variables X := {x : A } where x {0, 1,..., T L +1} deotes the startig time for 3
4 evet E i the schedule of R which is a required resource for the evet. If x = 0, the R is choosig ot to schedule E. We the costruct a DCOP with the variable set X := =1 K X. Let us ow defie a set X := {xm X : m = } X which is the collectio of variables pertaiig to the th resource. Clearly, X > 0, otherwise the resource is ot required i ay evet. If X = 1, let X := X {x} 0 where xi 0 0 is a dummy variable. Otherwise, X := X. The DCOP partitios the variables i X to a aget represetig the th resource s iterests. Let all the variables withi X itraaget lis) be fully coected. The additio of the dummy variable to sets X with cardiality oe is to esure that itraaget lis exist for all agets. Iteraget lis exist betwee the variables for all participats of a give evet, i.e., all the variables i X are fully coected. Propositio 3 The DCOP formulatio with private evets as variables, where the costrait betwee the variables x 1 ad x 2 2 whe x 1 = t 1 ad x 2 2 = t 2 taes o the utility f 1,, t 1 ; 2, 2, t 2 ) = MI {1 2 }I {1 = 2 }I {t1 t 2 } + I {1 = 2 }I {1 2 } f iter 1 ;, t 1 ; 2, t 2 ). 4) where f iter ;, t 1 ; 2, t 2 ) = M t 1 0, t 2 0, t 1 t 2 t 1 + L M t 1 0, t 2 0, t 2 t 1 t 2 + L 2 g;, t 1 ; 2, t 2 ) otherwise ad g;, t 1 ; 2, t 2 ) = 1 Z t 1 ) + Z 2 t 2 ) ) where Z i t i ) = X 1 L i l=1 i 0 t i + l 1) ) I {ti 0} with M > NT max where N is the umber of participats, T is the umber of time slots ad max = max,, yields the optimal solutio to the Distributed MultiEvet Schedulig DiMES) problem. Proof. The first term i 4) characterizes that a pealty of M is assessed o a iteraget li 1 2 ) for a commo evet = 2 ) for which the same startig time is ot selected t 1 t 2 ) by the coected resources. The latter term i 4) addresses itraaget costraits 1 = 2 ) betwee differet evets 2 ) where the li utility f iter ) esures that a pealty is icurred o a itraaget costrait if a schedulig coflict is created. Otherwise, the utility gai for a resource assigig a viable time for a evet is uiformly distributed amog the outgoig itraaget lis as deoted i g ). Let us assume that a pealty is icurred o a iteraget costrait. This implies that the required resources for a particular evet could ot agree o a commo time to start. Sice the total utility gai excludig pealties) for holdig a evet E caot exceed L max NT max < M, A t=1 there exists a solutio for the DCOP where the evet is ot scheduled which is at least as good as that with the evet scheduled. Let us ow assume that a pealty is icurred o a itraaget li. This implies that a aget has chose startig times for two evets that causes the same time slot to be assiged to two evets. By similar logic, the pealty M is sufficietly large such that by choosig ot to schedule oe of the evets ad allowig all other agets to choose ot to schedule that evet thereby avoidig icurrig a iteraget pealty), we obtai a higher quality solutio. The above aalysis implies that the optimal DCOP solutio is void of assigmets that would activate a pealty. Thus, x = x m, m, A. Give E ad some A, let us defie S E ) = if x = 0, ad S E ) = {x,..., x + L 1} if x 0. The, we have S E ) S E 2 ) =, 2 {1,..., K}, 2, A A 2. 5) 4
5 Otherwise, a pealty would have bee assessed. The global utility is the the sum of all itraaget lis devoid of pealties g )), which ca be represeted as N K =1 =1 = I { A } K L =1 A l=1 1 K X 1 Z x) =1 I { A } 0 x + l 1) ) I {x 0} = N K 1 = I { A }Zx ) K =1 =1 =1 A t S E ) 0 t) ). The solutio to the DCOP maximizes the previous expressio, which whe coupled with the o coflict coditio i 5) is idetical to the DiMES problem. 5
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationMultiplexers and Demultiplexers
I this lesso, you will lear about: Multiplexers ad Demultiplexers 1. Multiplexers 2. Combiatioal circuit implemetatio with multiplexers 3. Demultiplexers 4. Some examples Multiplexer A Multiplexer (see
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationCOMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS 2 CONTROL CHART FOR THE CHANGES IN A PROCESS
COMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 200617 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät LudwigMaximiliasUiversität Müche Olie at http://epub.ub.uimueche.de/940/
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationDesigning Incentives for Online Question and Answer Forums
Desigig Icetives for Olie Questio ad Aswer Forums Shaili Jai School of Egieerig ad Applied Scieces Harvard Uiversity Cambridge, MA 0238 USA shailij@eecs.harvard.edu Yilig Che School of Egieerig ad Applied
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More information1. Introduction. Scheduling Theory
. Itroductio. Itroductio As a idepedet brach of Operatioal Research, Schedulig Theory appeared i the begiig of the 50s. I additio to computer systems ad maufacturig, schedulig theory ca be applied to may
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationhp calculators HP 12C Statistics  average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics  average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationDynamic House Allocation
Dyamic House Allocatio Sujit Gujar 1 ad James Zou 2 ad David C. Parkes 3 Abstract. We study a dyamic variat o the house allocatio problem. Each aget ows a distict object (a house) ad is able to trade its
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationUniversal coding for classes of sources
Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametric
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationReview: Classification Outline
Data Miig CS 341, Sprig 2007 Decisio Trees Neural etworks Review: Lecture 6: Classificatio issues, regressio, bayesia classificatio Pretice Hall 2 Data Miig Core Techiques Classificatio Clusterig Associatio
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationOn the Capacity of Hybrid Wireless Networks
O the Capacity of Hybrid ireless Networks Beyua Liu,ZheLiu +,DoTowsley Departmet of Computer Sciece Uiversity of Massachusetts Amherst, MA 0002 + IBM T.J. atso Research Ceter P.O. Box 704 Yorktow Heights,
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationInference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval
Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT  Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationSequences and Series Using the TI89 Calculator
RIT Calculator Site Sequeces ad Series Usig the TI89 Calculator Norecursively Defied Sequeces A orecursively defied sequece is oe i which the formula for the terms of the sequece is give explicitly. For
More informationG r a d e. 2 M a t h e M a t i c s. statistics and Probability
G r a d e 2 M a t h e M a t i c s statistics ad Probability Grade 2: Statistics (Data Aalysis) (2.SP.1, 2.SP.2) edurig uderstadigs: data ca be collected ad orgaized i a variety of ways. data ca be used
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationCS100: Introduction to Computer Science
Review: History of Computers CS100: Itroductio to Computer Sciece Maiframes Miicomputers Lecture 2: Data Storage  Bits, their storage ad mai memory Persoal Computers & Workstatios Review: The Role of
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationFactors of sums of powers of binomial coefficients
ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationOnesample test of proportions
Oesample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:
More informationGroups of diverse problem solvers can outperform groups of highability problem solvers
Groups of diverse problem solvers ca outperform groups of highability problem solvers Lu Hog ad Scott E. Page Michiga Busiess School ad Complex Systems, Uiversity of Michiga, A Arbor, MI 481091234; ad
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationBond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond
What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixedicome security that typically pays periodic coupo paymets, ad a pricipal
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationA Recursive Formula for Moments of a Binomial Distribution
A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationNr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996
Nr 2 Iterpolatio of Discout Factors Heiz Cremers Willi Schwarz Mai 1996 Autore: Herausgeber: Prof Dr Heiz Cremers Quatitative Methode ud Spezielle Bakbetriebslehre Hochschule für Bakwirtschaft Dr Willi
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationCapacity of Wireless Networks with Heterogeneous Traffic
Capacity of Wireless Networks with Heterogeeous Traffic Migyue Ji, Zheg Wag, Hamid R. Sadjadpour, J.J. GarciaLuaAceves Departmet of Electrical Egieerig ad Computer Egieerig Uiversity of Califoria, Sata
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationMATH 083 Final Exam Review
MATH 08 Fial Eam Review Completig the problems i this review will greatly prepare you for the fial eam Calculator use is ot required, but you are permitted to use a calculator durig the fial eam period
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationFirewall Modules and Modular Firewalls
Firewall Modules ad Modular Firewalls H. B. Acharya Uiversity of Texas at Austi acharya@cs.utexas.edu Aditya Joshi Uiversity of Texas at Austi adityaj@cs.utexas.edu M. G. Gouda Natioal Sciece Foudatio
More informationMTOMTS Production Systems in Supply Chains
NSF GRANT #0092854 NSF PROGRAM NAME: MES/OR MTOMTS Productio Systems i Supply Chais Philip M. Kamisky Uiversity of Califoria, Berkeley Our Kaya Uiversity of Califoria, Berkeley Abstract: Icreasig cost
More informationTrading the randomness  Designing an optimal trading strategy under a drifted random walk price model
Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lieup for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More informationFOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More informationFloating Codes for Joint Information Storage in Write Asymmetric Memories
Floatig Codes for Joit Iformatio Storage i Write Asymmetric Memories Axiao (Adrew Jiag Computer Sciece Departmet Texas A&M Uiversity College Statio, TX 77843311 ajiag@cs.tamu.edu Vaske Bohossia Electrical
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More informationOn secure and reliable communications in wireless sensor networks: Towards kconnectivity under a random pairwise key predistribution scheme
O secure ad reliable commuicatios i wireless sesor etworks: Towards kcoectivity uder a radom pairwise key predistributio scheme Faruk Yavuz Dept. of ECE ad CyLab Caregie Mello Uiversity Moffett Field,
More information5.3. Generalized Permutations and Combinations
53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible
More informationStock Market Trading via Stochastic Network Optimization
PROC. IEEE CONFERENCE ON DECISION AND CONTROL (CDC), ATLANTA, GA, DEC. 2010 1 Stock Market Tradig via Stochastic Network Optimizatio Michael J. Neely Uiversity of Souther Califoria http://wwwrcf.usc.edu/
More information