Société de Calcul Mathématique, S. A. Algorithmes et Optimisation

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1 Société de Calcul Mathématique S A Algorithme et Optimiatio Radom amplig of proportio Berard Beauzamy Jue 2008 From time to time we fid a problem i which we do ot deal with value but with proportio For itace : A total budget ha to be pet ad we wat to tudy the repartitio betwee everal good ; A certai amout of pollutio i detected ad we wat to tudy the repartitio betwee everal factor I order to fid iteretig or dagerou ituatio oe uually trie to imulate variou poibilitie Sice o iformatio i ow a priori about the factor the law ued for uch a imulatio hould be a uiform law Decriptio of the problem Mathematically peaig thi mea : Simulate radom variable uch that : each 0 ad the law of the vector i the uiform law o the et : C x x ; x 0 x Uiform law mea that for a ample the umber of time we fall ito a ubet A C deped oly o the meaure of A It mea alo that the oit deity of the vector i cotat i the et C that i doe ot deped o the particular value of x x i C Siège ocial et bureaux : Faubourg Sait Hooré Pari Tel : Fax : Société Aoyme au capital de Euro RCS : Pari B SIRET : APE : 729Z

2 The dicrete verio of the uiform law i much eaier to udertad Fix a deomiator ay M We wat poitive iteger uch that M o our proportio are ad we wat that all repartitio have the ame probability M M idepedetly of the value of For itace tae 3 M 4 We have the poibilitie : ad each of thee 5 poibilitie hould have probability /5 2 Warig We have the ame warig a previouly : to geerate uiform variable ad the replace them by o 0 i wrog becaue we do ot obtai thi way a uiform law o C Thi i clear o the followig example (dimeio : A P P2 C Q Q2 O B 2

3 Aume we tae radom variable ( with uiform law o the egmet 0 The the umber of time the poit with coordiate ( fall ito the triagle OPP 2 i proportioal to the area of that triagle But the ormalized poit 2 i i the egmet Q Q2 if ad oly if the poit ( i i the 2 2 triagle OPP 2 But the area of the triagle deped upo it orietatio : for give legth of Q Q2 we have a larger triagle if it cotai the poit C ad maller at the extremitie whe the triagle cotai A or B So thi cotructio doe ot lead to a uiform law o the egmet AB 3 Geeratio of uiform variable o C The correct cotructio i a follow We follow the boo "No Uiform Radom Variate Geeratio" by Luc Devroye chapter 5 theorem 22 Theorem - Let be idepedet radom variable with expoetial law The the variable Y Y defied by : Y Y are poitive have um ad follow a uiform law o C Note : i order to geerate radom variable with expoetial law o hould geerate radom variable with uiform law Y ad the tae Log Y So aume you wat to geerate a ample of N 000 proportio for 40 good do the followig : for to N for to geerate tae Log for Y compute Y Y idepedet variable with uiform law o 0 replace each ext ext by 3

4 Proof of the Theorem Let S to implify the otatio Let ad Z Let ZS ( ) computed a a coditioal probability owig Z that i : f z be the deity of the oit law of the uple Z S S Z Z z x x ZS Thi deity ca be f ( z ) f ( ) f ( z) () where fs Z() deote the coditioal probability deity of S owig Z ad fz () z i the deity of Z Sice the variable are idepedet ad follow a expoetial law we have : f () z e Z x x ( The law of S owig x x i eay to fid Ideed P S a x x P a x x x x ad therefore : fs Z() e x x (3) for x x 0 otherwie We deduce from () ( (3) that : fzs ( z ) e (4) for x x 0 otherwie Now we compute the oit deity of variable i the deity (4) We et : S Thi i obtaied by a chage of S S Let U S S x x y y ; we get : fus ( u ) e (5) The deity of U ca be obtaied from the above formula itegratig with repect to We get : 4

5 f ( u) f ( u ) d e d! U (6) U S 0 0 ad we ee that thi value i cotat o the whole et x 0 x 0 x x But ow ice the deity of S S i cotat o i the deity of S S S S S S S (7) Thi coclude the proof of the Theorem Remar A pacage i Matlab due to Roger Stafford geerate radom umber with uiform law ad fixed um : it ca be foud o the Itered uder the ame of radfixedumm Thi pacage ue aother approach more complicated tha the approach decribed here (uig expoetial variable) but it ca olve more geeral problem : fid x i with a xi b ad xi S i Acowledgemet We tha Paul Deheuvel for idicatig the boo by Luc Devroye Luc Devroye for hi commet about the proof ad Roger Stafford for hi commet about the Matlab pacage 5

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