A Derivation of Bill James Pythagorean Won-Loss Formula

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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) )

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