# A Derivation of Bill James Pythagorean Won-Loss Formula

Save this PDF as:

Size: px
Start display at page:

Download "A Derivation of Bill James Pythagorean Won-Loss Formula"

## Transcription

1 A Drivation of Bill Jams Pythagoran Won-Loss Formula Ths nots wr compild by John Paul Cook from a papr by Dr. Stphn J. Millr, an Assistant Profssor of Mathmatics at Williams Collg, for a talk givn to th Applid Mathmatics Sminar at th Univrsity of Oklahoma on April 7, Th Pythagoran Formula Sabrmtrician Bill Jams drivd th Pythagoran W-L Formula to srv as an xpctd valu for a basball tam s winning prcntag, basd on th numbr of runs a tam scors and allows ovr th cours of a sason: Runs Scord 2 Expctd Win Prcntag Runs Scord 2 + Runs Allowd 2 By Jams own admission, his own drivation of th formula, whil th rsult of data analysis, had littl basis in statistical and mathmatical thory. W will show that, undr rasonabl statistical assumptions, Jams formula dos, in fact, follow mathmatically. Exampl.. Th application of this formula: ) On August 4, 2008 th Txas Rangrs had scord 640 runs and allowd 667, lading to a Pythagoran W-L% of ). Thir actual rcord at th tim was ). Th Rangrs ndd th sason at ). 2) On July 20, 2008 th Clvland Indians had scord 443 runs and allowd 434, lading to a Pythagoran W-L% of ). Thir actual rcord at th tim was ). Th Indians ndd th sason at ). 2. Assumptions and th Wibull Distribution Rmark 2.. W will b making th following assumptions: ) Runs scord and runs allowd can b modld by continuous random variabls This allows us to rplac discrt sums with continuous intgrals, which ar much asir to solv and much nicr to work with. Whil continuous run distributions do not mak sns, w hop that this computationally usful assumption rasonably approximats th corrsponding discrt distribution.

2 2 2) Runs scord and runs allowd can b modld by thr-paramtr Wibull distributions Wibull random variabls hav flxibl shap paramtrs which mak it asir to fit to obsrvd data than bttr known distributions, such as th xponntial. Th xponntial distribution dcays too slowly to b ralistic, lading to too many gams with absurdly high scors, whil th dcay of th Wibull distribution is much mor ralistic. 3) Runs scord and runs allowd pr gam ar statistically indpndnt Whil thy can not b ntirly indpndnt as gams can not nd in tis), modifid Chi-Squard tsts hav shown that runs scord and runs allowd pr gam ar statistically indpndnt. Dfinition 2.2. Th probability dnsity function for a thr-paramtr Wibull distribution is givn by: ) β ) fx; α, β, ) xβ α ) if x β, 0 othrwis α α Lmma 2.3. Th xpctd valu for a thr-paramtr Wibull random variabl X is givn by whr Γ is th Γ-function Γs) 0 EX) αγ + ) + β u u s du for s {z C : Rz) > 0} Exampl 2.4. Using th Wibull distribution: Suppos that th Txas Rangrs runs scord pr gam is givn by a Wibull distribution with paramtrs α, β, ) 5, 0, 2). Thn th probability that a tam scors btwn and 3 runs in a givn gam is 3 2 ) ) x Rmark 2.5. Som nots on th paramtrs: ) is a gnralizd shap paramtr. Whn w hav th xponntial distribution, and whn 2 w hav th Rayligh distribution. Th paramtr will nd up bing th xponnt in th final formula. Whil Bill Jams usd 2 most likly for simplicity), subsqunt analysis of Jams formula has shown a mor accurat xponnt to b about.82. W gnraliz

3 this paramtr to lav opn th possibility to find a bst-fit xponnt for whatvr lagu is bing analyzd. Analysis of th data has shown that w can gt good fits for runs scord and runs allowd by using th sam for both runs scord and runs allowd. 2) β is incorporatd to partially account for our us of continuous random variabls instad of discrt random variabls. It will also hlp adjust this formula to othr sports by allowing us to xamin scors abov a baslin for xampl, no basktball tam scors fwr than 25 points in a gam). Incorporation of this paramtr will giv us a final rsult of: RS β) Expctd Win Prcntag RS β) + RA β) W us th sam β for both runs scord and runs allowd bcaus adding th sam numbr to both runs scord and runs allowd will not chang th outcom of th gam, and thus prsrvs th xpctd winning prcntag. 3) Rcall from abov that if X is a Wibull RV, EX) αγ + ) + β Sinc w ar using th sam and β for runs scord and runs allowd, w will us distinct α for both which w dnot and ). In fact, w will choos and such that th random variabls for runs scord and runs allowd conform to th abov xpctd valu formula. W ar almost rady to prov our main rsult, but w nd a fw mor prliminaris. Rcall that our goal is to calculat th projctd winning prcntag of a basball tam. In othr words, th probability that, in any givn gam, th tam in qustion will scor mor runs than it allows. Sinc this probability dpnds jointly on our two Wibull) random variabls for runs scord and runs allowd, w nd to us a joint probability dnsity function: Dfinition 2.6. Suppos that X and Y ar two continuous random variabls dfind ovr a givn sampl spac S. Th joint probability dnsity function of X and Y, f X,Y x, y), is th surfac having th proprty that for any rgion R in th xy-plan, P X, Y R) f X,Y x, y) dy R 3

4 4 Additionally, as w discussd arlir, w ar assuming runs scord and runs allowd to b statistically indpndnt: Dfinition 2.7. Two random variabls X and Y ar statistically) indpndnt if and only if f X,Y x, y) f X x)f Y y) for all x, y. Rmark 2.8. Also rcall that th intgral ovr all possibl valus of a probability dnsity function is : W ar now rady to procd. f X x) 3. Proof of Bill Jams Pythagoran Won-Loss Formula Thorm 3.. Lt runs scord and runs allowd b indpndnt thrparamtr Wibull random variabls dnotd by X,, β, ) and Y,, β, ), rspctivly. Suppos and ar chosn such that EX) Γ + ) + β and EY ) Γ + ) + β. Put EX) RS and EY ) RA. If > 0, thn Expctd Win Prcntag P X > Y ) Proof. First of all, w hav that RS β) RS β) + RA β) RS Γ + ) + β and RA Γ + ) + β So that RS β Γ + ) and RA β Γ + )

5 Now P X > Y ) xβ yβ xβ yβ x0 x0 x0 x0 y0 f X,Y x, y) dy β ) ) ) x/ ) xβ ) xβ ) x y0 yβ f X x)f Y y) dy y β ) y ) ) yβ ) y y ) y/ ) ] yx ) [ x/ ) y/) y0 ) [ ] x/ ) x/) ) x/ ) ) x x0 Lt α givs us: x0 α RS + α RA α RS + α RS + α RS α RA x0 α RS )) + α RA ) x α RS dy 5 dy dy )) + α RA. Thn multiplying th prvious xprssion by α α x ) x α) α α α RS + RS β ) Γ + ) RS β ) RA β + Γ + ) Γ + ) RS β) RS β) + RA β) )

### The Matrix Exponential

Th Matrix Exponntial (with xrciss) 92.222 - Linar Algbra II - Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial

More information

### Non-Homogeneous Systems, Euler s Method, and Exponential Matrix

Non-Homognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous first-ordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach

More information

### Exponential Growth and Decay; Modeling Data

Exponntial Growth and Dcay; Modling Data In this sction, w will study som of th applications of xponntial and logarithmic functions. Logarithms wr invntd by John Napir. Originally, thy wr usd to liminat

More information

### A Note on Approximating. the Normal Distribution Function

Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and

More information

### (Analytic Formula for the European Normal Black Scholes Formula)

(Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually

More information

### 14.3 Area Between Curves

14. Ara Btwn Curvs Qustion 1: How is th ara btwn two functions calculatd? Qustion : What ar consumrs and producrs surplus? Earlir in this chaptr, w usd dfinit intgrals to find th ara undr a function and

More information

### Module 7: Discrete State Space Models Lecture Note 3

Modul 7: Discrt Stat Spac Modls Lctur Not 3 1 Charactristic Equation, ignvalus and ign vctors For a discrt stat spac modl, th charactristic quation is dfind as zi A 0 Th roots of th charactristic quation

More information

### Poisson Distribution. Poisson Distribution Example

Poisson Distribution Can b usd to valuat th probability of an isolatd vnt occurring a spcific numbr of tims in a givn tim intrval,.g. # of faults, # of lightning stroks tim intrval Rquirmnts: -Evnts must

More information

### New Basis Functions. Section 8. Complex Fourier Series

Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ral-valud Fourir sris is xplaind and formula ar givn for convrting

More information

### The Normal Distribution: A derivation from basic principles

Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn

More information

### e = C / electron Q = Ne

Physics 0 Modul 01 Homwork 1. A glass rod that has bn chargd to +15.0 nc touchs a mtal sphr. Aftrword, th rod's charg is +8.00 nc. What kind of chargd particl was transfrrd btwn th rod and th sphr, and

More information

### Traffic Flow Analysis (2)

Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,

More information

### CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions

CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:

More information

### Chi-Square. Hypothesis: There is an equal chance of flipping heads or tails on a coin. Coin A. Expected (e) (o e) (o e) 2 (o e) 2 e

Why? Chi-Squar How do you know if your data is th rsult of random chanc or nvironmntal factors? Biologists and othr scintists us rlationships thy hav discovrd in th lab to prdict vnts that might happn

More information

### Genetic Drift and Gene Flow Illustration

Gntic Drift and Gn Flow Illustration This is a mor dtaild dscription of Activity Ida 4, Chaptr 3, If Not Rac, How do W Explain Biological Diffrncs? in: How Ral is Rac? A Sourcbook on Rac, Cultur, and Biology.

More information

### Question 3: How do you find the relative extrema of a function?

ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating

More information

### 7 Timetable test 1 The Combing Chart

7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing

More information

### Section 7.4: Exponential Growth and Decay

1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart

More information

### Adverse Selection and Moral Hazard in a Model With 2 States of the World

Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,

More information

### Factorials! Stirling s formula

Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical

More information

### Econ 371: Answer Key for Problem Set 1 (Chapter 12-13)

con 37: Answr Ky for Problm St (Chaptr 2-3) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc

More information

### 10/06/08 1. Aside: The following is an on-line analytical system that portrays the thermodynamic properties of water vapor and many other gases.

10/06/08 1 5. Th watr-air htrognous systm Asid: Th following is an on-lin analytical systm that portrays th thrmodynamic proprtis of watr vapor and many othr gass. http://wbbook.nist.gov/chmistry/fluid/

More information

### Simulated Radioactive Decay Using Dice Nuclei

Purpos: In a radioactiv sourc containing a vry larg numbr of radioactiv nucli, it is not possibl to prdict whn any on of th nucli will dcay. Although th dcay tim for any on particular nuclus cannot b prdictd,

More information

### by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia

Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs

More information

### Solutions to Homework 8 chem 344 Sp 2014

1. Solutions to Homwork 8 chm 44 Sp 14 .. 4. All diffrnt orbitals mans thy could all b paralll spins 5. Sinc lctrons ar in diffrnt orbitals any combination is possibl paird or unpaird spins 6. Equivalnt

More information

### Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY

Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl

More information

### Inference by Variable Elimination

Chaptr 5 Infrnc by Variabl Elimination Our purpos in this chaptr is to prsnt on of th simplst mthods for gnral infrnc in Baysian ntworks, known as th mthod of Variabl Elimination. 5.1 Introduction Considr

More information

### Category 7: Employee Commuting

7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil

More information

### Principles of Humidity Dalton s law

Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

More information

### Ground Fault Current Distribution on Overhead Transmission Lines

FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 19, April 2006, 71-84 Ground Fault Currnt Distribution on Ovrhad Transmission Lins Maria Vintan and Adrian Buta Abstract: Whn a ground fault occurs on

More information

### ME 612 Metal Forming and Theory of Plasticity. 6. Strain

Mtal Forming and Thory of Plasticity -mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.

More information

### Upper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing

Uppr Bounding th Pric of Anarchy in Atomic Splittabl Slfish Routing Kamyar Khodamoradi 1, Mhrdad Mahdavi, and Mohammad Ghodsi 3 1 Sharif Univrsity of Tchnology, Thran, Iran, khodamoradi@c.sharif.du Sharif

More information

### Variance Components (VARCOMP)

Varianc Componnts (VARCOP) Notation Th Varianc Componnts procdur provids stimats for variancs of random ffcts undr a gnral linar modl framwor. Four typs of stimation mthods ar availabl in this procdur.

More information

### QUANTITATIVE METHODS CLASSES WEEK SEVEN

QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.

More information

### Some Useful Integrals of Exponential Functions

prvious indx nxt Som Usful Intgrls of Exponntil Functions Michl Fowlr W v shown tht diffrntiting th xponntil function just multiplis it by th constnt in th xponnt, tht is to sy, d x x Intgrting th xponntil

More information

### Statistical Machine Translation

Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm

More information

### Performance Evaluation

Prformanc Evaluation ( ) Contnts lists availabl at ScincDirct Prformanc Evaluation journal hompag: www.lsvir.com/locat/pva Modling Bay-lik rputation systms: Analysis, charactrization and insuranc mchanism

More information

### AP Calculus AB 2008 Scoring Guidelines

AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity.

More information

### Gas Radiation. MEL 725 Power-Plant Steam Generators (3-0-0) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi

Gas Radiation ME 725 Powr-Plant Stam Gnrators (3-0-0) Dr. Prabal Talukdar Assistant Profssor Dpartmnt of Mchanical Enginring T Dlhi Radiation in absorbing-mitting mdia Whn a mdium is transparnt to radiation,

More information

### Modern Portfolio Theory (MPT) Statistics

Modrn Portfolio Thory (MPT) Statistics Morningstar Mthodology Papr May 9, 009 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription

More information

### The Neolithic transition, a major episode in human history, is

Synthsis btwn dmic and cultural diffusion in th Nolithic transition in Europ Joaquim Fort 1 Complx Systms Laboratory, Dpartmnt of hysics, Univrsity of Girona, ES-1771 Girona, Catalonia, Spain Editd by

More information

### SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* Rostov-on-Don. Russia

SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* Rostov-on-Don. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs

More information

### The example is taken from Sect. 1.2 of Vol. 1 of the CPN book.

Rsourc Allocation Abstract This is a small toy xampl which is wll-suitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of C-nts. Hnc, it can b rad by popl

More information

### The Constrained Ski-Rental Problem and its Application to Online Cloud Cost Optimization

3 Procdings IEEE INFOCOM Th Constraind Ski-Rntal Problm and its Application to Onlin Cloud Cost Optimization Ali Khanafr, Murali Kodialam, and Krishna P. N. Puttaswam Coordinatd Scinc Laborator, Univrsit

More information

### Deer: Predation or Starvation

: Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,

More information

### Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s1660-015-0741-1 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans Wi-Mao Qian

More information

### http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force

ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd

More information

### Foreign Exchange Markets and Exchange Rates

Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls

More information

### On the moments of the aggregate discounted claims with dependence introduced by a FGM copula

On th momnts of th aggrgat discountd claims with dpndnc introducd by a FGM copula - Mathiu BARGES Univrsité Lyon, Laboratoir SAF, Univrsité Laval - Hélèn COSSETTE Ecol Actuariat, Univrsité Laval, Québc,

More information

### Singleton Theorem Using Models

Singlton Thorm Using Modls Srivathsan B, Igor Walukiwicz LaBRI Paris, March 2010 Srivathsan B, Igor Walukiwicz (LaBRI) Singlton Thorm Using Modls Paris, March 2010 1 / 17 Introduction Singlton Thorm [Statman

More information

### CHAPTER EIGHT. Making use of the limit formula developed in Chapter 1, it can be shown that

CONTINUOUS COMPOUNDING CHAPTER EIGHT In prvious stions, th as of m ompoundings pr yar was disussd In th as of ontinuous ompounding, m is allowd to tnd to infinity Th mathmatial rlationships ar fairly asy

More information

### [ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2)

MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (volts-sc/rad Motor torqu constant (lb-in/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris

More information

### L13: Spectrum estimation nonparametric and parametric

L13: Spctrum stimation nonparamtric and paramtric Lnnart Svnsson Dpartmnt of Signals and Systms Chalmrs Univrsity of Tchnology Problm formulation Larning objctivs Aftr today s lctur you should b abl to

More information

### Lecture 3: Diffusion: Fick s first law

Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th

More information

### Parallel and Distributed Programming. Performance Metrics

Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:

More information

### Downlink Coverage Probability in MIMO HetNets with Flexible Cell Selection

Downlin Covrag robability in MIMO HtNts with Flxibl Cll Slction Abhish K. Gupta, Harprt S. Dhillon, Sriram Vishwanath, Jffry G. Andrws Abstract In this papr, w study th covrag probability of a K-tir multipl-input

More information

### Financial Mathematics

Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution

More information

### ON THE KAUFFMAN BRACKET OF PRETZEL LINKS. The University of Liverpool

ON THE KAUFFMAN BRACKET OF PRETZEL LNKS Pdro María Gonzálz Manchón Th Univrsity of Livrool manchon@trras manchon@livacuk ABSTRACT n this not w giv a closd-form formula for th Kauffman brackt of rtzl links

More information

### CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.

Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by

More information

### Externalities in Online Advertising

Extrnalitis in Onlin Advrtising Arpita Ghosh Yahoo! Rsarch 2821 Mission Collg Blvd. Santa Clara, CA 95054 arpita@yahoo-inc.com Mohammad Mahdian Yahoo! Rsarch 2821 Mission Collg Blvd. Santa Clara, CA 95054

More information

### Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails

More information

### 5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power

Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim

More information

### Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental

Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal Zvi Lotkr zvilo@cs.bgu.ac.il Dpt. of Comm. Systms Enginring Bn Gurion Univrsity Br Shva Isral Boaz Patt-Shamir Dror Rawitz {boaz, rawitz}@ng.tau.ac.il

More information

### Notes on Fermi-Dirac Integrals 3 rd Edition

Nots on rmi-dirac Intgrals 3 rd Edition Rasong Kim and Mark Lundstrom Ntwork for Computational Nanotchnology Purdu Univrsity Dcmbr 1, 8 (Last rvisd on August 4, 11) 1. Introduction rmi-dirac intgrals appar

More information

### MAXIMAL CHAINS IN THE TURING DEGREES

MAXIMAL CHAINS IN THE TURING DEGREES C. T. CHONG AND LIANG YU Abstract. W study th problm of xistnc of maximal chains in th Turing dgrs. W show that:. ZF + DC+ Thr xists no maximal chain in th Turing dgrs

More information

### SAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII

SAMPLE QUESTION PAPER MATHEMATICS (4) CLASS XII 6-7 Tim allowd : 3 hours Maimum Marks : Gnral Instructions: (i) All qustions ar compulsor. (ii) This qustion papr contains 9 qustions. (iii) Qustion - 4

More information

### Incomplete 2-Port Vector Network Analyzer Calibration Methods

Incomplt -Port Vctor Ntwork nalyzr Calibration Mthods. Hnz, N. Tmpon, G. Monastrios, H. ilva 4 RF Mtrology Laboratory Instituto Nacional d Tcnología Industrial (INTI) Bunos irs, rgntina ahnz@inti.gov.ar

More information

### SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM

RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Ths paprs

More information

### SPECIAL VOWEL SOUNDS

SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)

More information

### Budget Optimization in Search-Based Advertising Auctions

Budgt Optimization in Sarch-Basd Advrtising Auctions ABSTRACT Jon Fldman Googl, Inc. Nw York, NY jonfld@googl.com Martin Pál Googl, Inc. Nw York, NY mpal@googl.com Intrnt sarch companis sll advrtismnt

More information

### Gold versus stock investment: An econometric analysis

Intrnational Journal of Dvlopmnt and Sustainability Onlin ISSN: 268-8662 www.isdsnt.com/ijds Volum Numbr, Jun 202, Pag -7 ISDS Articl ID: IJDS20300 Gold vrsus stock invstmnt: An conomtric analysis Martin

More information

### Mark B. Garman and Michael J. Klass 1

Mark B. Garman and Michal J. Klass Univrsity of California, Brkly On th Estimation of Scurity Pric Volatility from Historical Data Abstract: Improvd stimators of scurity pric volatility ar formulatd. Ths

More information

### intro Imagine that someone asked you to describe church using only the bible. What would you say to them?

intro Imagin that somon askd you to dscrib church using only th bibl. What would you say to thm? So many of th things w'v mad church to b arn't ssntial in scriptur. W'r on a journy of r-imagining what

More information

### Intermediate Macroeconomic Theory / Macroeconomic Analysis (ECON 3560/5040) Final Exam (Answers)

Intrmdiat Macroconomic Thory / Macroconomic Analysis (ECON 3560/5040) Final Exam (Answrs) Part A (5 points) Stat whthr you think ach of th following qustions is tru (T), fals (F), or uncrtain (U) and brifly

More information

### CALCULATING MARGINAL PROBABILITIES IN PROC PROBIT Guy Pascale, Memorial Health Alliance

CALCULATING MARGINAL PROBABILITIES IN PROC PROBIT Guy Pascal, Mmorial Halth Allianc Introduction Th PROBIT procdur within th SAS systm provids a simpl mthod for stimating discrt choic variabls (i.. dichotomous

More information

### PES 3950/PHYS 6950: Homework Assignment 5

PES 3950/PHYS 6950: Homwork ssignmnt 5 Handd out: Wdnsday pril 8 Du in: Wdnsday pril, at t start of class at 3:05 pm sarp Sow all working and rasoning to rciv full points. Qustion [5 points] a)[0 points]

More information

### Sigmoid Functions and Their Usage in Artificial Neural Networks

Sigmoid Functions and Thir Usag in Artificial Nural Ntworks Taskin Kocak School of Elctrical Enginring and Computr Scinc Applications of Calculus II: Invrs Functions Eampl problm Calculus Topic: Invrs

More information

### The price of liquidity in constant leverage strategies. Marcos Escobar, Andreas Kiechle, Luis Seco and Rudi Zagst

RACSAM Rv. R. Acad. Cin. Sri A. Mat. VO. 103 2, 2009, pp. 373 385 Matmática Aplicada / Applid Mathmatics Th pric of liquidity in constant lvrag stratgis Marcos Escobar, Andras Kichl, uis Sco and Rudi Zagst

More information

### 14.02 Principles of Macroeconomics Problem Set 4 Solutions Fall 2004

art I. Tru/Fals/Uncrtain Justify your answr with a short argumnt. 4.02 rincipls of Macroconomics roblm St 4 Solutions Fall 2004. High unmploymnt implis that th labor markt is sclrotic. Uncrtain. Th unmploymnt

More information

### Electronic Commerce. and. Competitive First-Degree Price Discrimination

Elctronic Commrc and Comptitiv First-Dgr Pric Discrimination David Ulph* and Nir Vulkan ** Fbruary 000 * ESRC Cntr for Economic arning and Social Evolution (ESE), Dpartmnt of Economics, Univrsity Collg

More information

### Introduction to Finite Element Modeling

Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation

More information

### 5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This

More information

### In the previous two chapters, we clarified what it means for a problem to be decidable or undecidable.

Chaptr 7 Computational Complxity 7.1 Th Class P In th prvious two chaptrs, w clarifid what it mans for a problm to b dcidabl or undcidabl. In principl, if a problm is dcidabl, thn thr is an algorithm (i..,

More information

### Expert-Mediated Search

Exprt-Mdiatd Sarch Mnal Chhabra Rnsslar Polytchnic Inst. Dpt. of Computr Scinc Troy, NY, USA chhabm@cs.rpi.du Sanmay Das Rnsslar Polytchnic Inst. Dpt. of Computr Scinc Troy, NY, USA sanmay@cs.rpi.du David

More information

### Long run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange

Lctur 6: Th Forign xchang Markt xchang Rats in th long run CON 34 Mony and Banking Profssor Yamin Ahmad xchang Rats in th Short Run Intrst Parity Big Concpts Long run: Law of on pric Purchasing Powr Parity

More information

### An International Journal of the Polish Statistical Association

STATISTICS IN TRANSITION nw sris An Intrnational Journal of th Polish Statistical Association CONTENTS From th Editor... Submission information for authors... 5 Sampling mthods and stimation CIEPIELA P.,

More information

### the so-called KOBOS system. 1 with the exception of a very small group of the most active stocks which also trade continuously through

Liquidity and Information-Basd Trading on th Ordr Drivn Capital Markt: Th Cas of th Pragu tock Exchang Libor 1ÀPH³HN Cntr for Economic Rsarch and Graduat Education, Charls Univrsity and Th Economic Institut

More information

### Job shop scheduling with unit processing times

Job shop schduling with unit procssing tims Nikhil Bansal Tracy Kimbrl Maxim Sviridnko Abstract W considr randomizd algorithms for th prmptiv job shop problm, or quivalntly, th cas in which all oprations

More information

### Improving Managerial Accounting and Calculation of Labor Costs in the Context of Using Standard Cost

Economy Transdisciplinarity Cognition www.ugb.ro/tc Vol. 16, Issu 1/2013 50-54 Improving Managrial Accounting and Calculation of Labor Costs in th Contxt of Using Standard Cost Lucian OCNEANU, Constantin

More information

### WORKLOAD STANDARD DEPARTMENT OF CIVIL ENGINEERING. for the. Workload Committee : P.N. Gaskin (Chair) J.W. Kamphuis K. Van Dalen. September 24, 1997

WORKLOAD STANDARD for th DEPARTMENT OF CIVIL ENGINEERING Sptmbr 24, 1997 Workload Committ : P.N. Gaskin (Chair) J.W. Kamphuis K. Van Daln 9 2 TABLE OF CONTENTS l. INTRODUCTION... 3 2. DEFINITION OF WORKLOAD

More information

### Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.

Vrsion.0 Gnral Crtificat of Education (A-lvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,

More information

### Logo Design/Development 1-on-1

Logo Dsign/Dvlopmnt 1-on-1 If your company is looking to mak an imprssion and grow in th marktplac, you ll nd a logo. Fortunatly, a good graphic dsignr can crat on for you. Whil th pric tags for thos famous

More information

### FACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data

FACULTY SALARIES FALL 2004 NKU CUPA Data Compard To Publishd National Data May 2005 Fall 2004 NKU Faculty Salaris Compard To Fall 2004 Publishd CUPA Data In th fall 2004 Northrn Kntucky Univrsity was among

More information

### Lagrangian Formulation of a Consistent Relativistic Guiding Center Theory

Lagrangian Formulation of a Consistnt Rlativistic Guiding Cntr Thory H. K. Wimml Max-Planck-Institut für Plasmaphysik. EURATOM Association, Garching bi Münchn Z. Naturforsch. 38 a, 601-607 (1983); rcivd

More information

### Finite Element Vibration Analysis

Finit Elmnt Vibration Analysis Introduction In prvious topics w larnd how to modl th dynamic bhavior of multi-dof systms, as wll as systms possssing infinit numbrs of DOF. As th radr may raliz, our discussion

More information

### Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D

24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd

More information

### The Fourier Transform

Th Fourir Transfor Larning outcos Us th Discrt Fourir Transfor to prfor frquncy analysis on a discrt (digital) signal Eplain th significanc of th Fast Fourir Transfor algorith; Eplain why windowing is

More information

### econstor Make Your Publication Visible

constor Mak Your Publication Visibl A Srvic of Wirtschaft Cntr zbwlibniz-informationszntrum Economics Rich, Robrt; Tracy, Josph Working Papr Th rlationship btwn xpctd inflation, disagrmnt, and uncrtainty:

More information

### Chapter 9 Nuclear Radiation Radioactive Isotopes 9.1 radioactive isotope Natural Radioactivity

Chaptr 9 Nuclar Radiation 9. Natural Radioactivity Radioactiv Isotops A radioactiv isotop has an unstabl nuclus. mits radiation to bcom mor stabl. can b on or mor of th isotops of an lmnt 2 Nuclar Radiation

More information

### Differential Equations (MTH401) Lecture That a non-homogeneous linear differential equation of order n is an equation of the form n

Diffrntial Equations (MTH40) Ltur 7 Mthod of Undtrmind Coffiints-Surosition Aroah Rall. That a non-homognous linar diffrntial quation of ordr n is an quation of th form n n d d d an + a a a0 g( ) n n +

More information