The Hypergeometric Coupon Collection Problem and its Dual

Transcription

1 Joural of Idustrial ad Systes Egieerig Vol., o., pp -7 Sprig 7 The Hypergeoetric Coupo Collectio Proble ad its Dual Sheldo M. Ross Epstei Departet of Idustrial ad Systes Egieerig, Uiversity of Souther Califoria, Los Ageles, CA, USA (soss@usc.edu) ABSTRACT Suppose a ur cotais M balls, of differet types, which are reoved fro the ur i a uifor rado aer. I the hypergeoetric coupo collectio proble, we are iterested i the set of balls that have bee reoved at the oet whe at least oe ball of each type has bee reoved. I its dual, we are iterested i the set of reoved balls at the first oet that this set cotais all of the balls of at least oe type.. ITRODUCTIO AD SUMMARY Cosider a ur that cotais M balls, of which i are of type i, for i,...,, M i i. The hypergeoetric coupo collectig proble arises whe balls are reoved fro the ur i uifor rado aer util at least oe ball of each type has bee reoved. Let i deote the uber of type i balls that are reoved i the coupo collectig proble, ad let i deote the total uber of reoved balls. I sectio we derive the oit probability ass fuctio of,..., as well as the argial ass fuctios ad the eas ad variaces of the i. We also give forulas for the factorial oets of. The tail distributio P (>r) is explicitly obtaied whe all i are equal, ad upper ad lower bouds are provided i the geeral case. We also show that P(>r) is a Schur covex fuctio of the paraeters,...,. The dual of the hypergeoetric coupo collectig proble, which stops reovig balls at the oet whe all the balls of at least oe type have bee reoved, is cosidered i Sectio 3.. RESULTS FOR THE HYPERGEOMETRIC COUPO COLLECTIO PROBLEM We start with the oit ass fuctio of,...,. Propositio With r i, P ( i, i,..., i) i : i M M r + ( r )

2 Ross Proof: With L beig the last of the types to have oe of its balls reoved, we have P ( i, i,, i ) P( i, i,, i, L ) : i i : i M M r + r To obtai the argial ass fuctio of we will fid it useful to cosider a cotiuous tie odel i which each of the balls is reoved at rado tie that is uiforly distributed o (, ), with these M ties beig idepedet. Clearly, the order i which the balls are reoved i this cotiuous tie odel is probabilistically the sae as i the origial odel. Propositio r r r, P( ) ( t) [ ( t) ] dt + ( t) t( t) [ ( t) ] dt i i i r, P( i) ( t) t ( t) [ ( t) ] dt, i > Proof: Let T deote the tie of the first reoval of type ball. The, with L desigatig the last type ball reoved P ( i) P ( il, ) + P ( il, ) P ( il, ) + P ( il, T t ) ( t) dt i i + i r, P ( il, ) ( t) t( t) [ ( t) ] dt where ( t) [ ( t) ] dt, if r, if i > P ( il, ) ad the proof is coplete. Rear: The precedig result ca be used to obtai a expressio for P ( i ), the expected uber of types to have exactly i balls reoved. Expressios for these quatities i the classical coupo collectig proble - where each ew coupo is, idepedetly of the past, of type i with probability p i, i,...,, - were give i Adler et al. (3). We ow obtai the ea ad variace of.

3 The Hypergeoetric Coupo Collectio 3 Propositio 3 With r, p ( t) ( ( t) ) dt+ ( t) t ( ( t) ) dt, E [ ] p ( ) + r, Var( ) p ( p ) [ ( t) t ( ( t) ) dt p ] Proof: uber the type balls, ad let R i deote the evet that type ball uber i is i the fial set, i,...,. The, PR ( ) PR (, L ) + PR (, L ) i i i r, ( t) ( ( t) ) dt+ ( t) t ( ( t) ) dt Also, if, the for i s PRR ( ) PRR (, L ) i s i s r, ( t) t ( ( t) ) dt the result follows sice I. Ri The ext propositio yields a forula for factorial oets of balls oe eeds to reove to obtai a coplete set., the total uber of Propositio 4 ( M + r)! i r E[ ( + )...( + r )] r [ ( p )]( p) dp M! Proof: Cosider a syste with M copoets i which each copoet is either worig or failed, ad suppose there is a ootoe structure fuctio that defies whe the syste wors as a fuctio of which copoets are worig. Suppose that the M iitially worig copoets fail i a rado, uiforly distributed order, ad let equal to the uber of copoets that eed fail to cause the syste to fail. It was show by Ross et al. (98) that ( M + r)! r E[ ( + )...( + r )] r r( p)( p) dp M! where r(p) is the probability that the syste wors whe each copoet idepedetly wors with probability p.

4 4 Ross For a syste of M copoets, partitioed ito disoit subsets of sizes,...,, that is said to wor if all of the copoets of at least oe subset wor r( p) ( p i ) dp ad the result follows. Our ext result shows that the distributio fuctio of is a Schur fuctio of the paraeters (,..., ). Propositio 5 P (>r) is a Schur covex fuctio of (,..., ). Proof: Let (,..., ) be the uber of balls reoved to obtai at least oe of each type. It suffices to show, for >, that P ( (,,,..., ) > r) P ( (, +,,..., ) > r) 3 3 Let I(,..., ) be the idicator variable of the evet that the set of the first balls reoved cotais at least oe of each of the types 3,...,; ad let R(,..., ) deote the total uber of balls of types 3,...,, that are aog the first r balls reoved. The P ( (,..., ) > r) P ( (,..., ) > ri (,..., ) ir, (,..., ) ) P( I(,..., ) i, R(,..., ) ) Because (I,R ) (I(,, 3,, ),R(,, 3,, )), ad (I,R ) (I( -, +, 3,, ),R( -, +, 3,, )) have the sae oit distributio, it suffices to show that for > ad > P( (,..., ) > ri, R r ) P( (, +,..., ) > ri, R r ) which is equivalet to showig that P ( (, ) > ) P ( (, + ) > ) or, equivaletly, that or, equivaletly, that

5 The Hypergeoetric Coupo Collectio 5 or, equivaletly, that ( )! ( )!( )! which is iediate. The followig propositio gives bouds for P( >r), as well as a closed for expressio whe i,,,. Propositio 6 M M r r M M P ( > r) M M r r + r r If all, i ( r) ( ) P i+ > i M i r M r Proof: Let A i deote the evet that there are o type i balls aog the first r selected. The P> r PA A A ( ) (... ) The right had iequality follows fro Boole s iequality, ad the left had oe fro the coditioal expectatio iequality (Ross, ). The result whe all, follows fro the iclusio-exclusio idetity. Rears: (a) The coditioal expectatio iequality, stroger tha the secod oet iequality, states that P( A ) PA ( ) + PAA ( ) i (b) Although we could also use the iclusio-exclusio idetity to derive a expressio for P( >r) i the geeral case, it would require suig over ters.

6 6 Ross (c) Whe all i,, we ca also efficietly copute P( >r) by a recursio. For, i <, let A(i,) deote a geeric rado variable havig the distributio of the uber of additioal balls oe ust reove to have coplete set if balls reai i the ur ad there are still i types that have yet to be collected. The, with we have the recursio P (, i ) P( A(, i ) ) i i P(, i ) P ( i, ) + P (, i ), i> with the boudary coditio P (, ) ; uerical coputatio yields the result P(>r)-P r (,). 3. THE DUAL PROBLEM The dual proble to the oe so far cosidered is iterested i the uber of balls that have bee reoved at the first oet that all the balls of ay of the types have bee reoved. ow, iagiig that we cotiue reovig balls util they are all tae fro the ur, the outcoe of the experiet for both the origial ad dual proble is a vector ( i,..., i M ), where i is the type of the th ball reoved. ow, as a fuctio of the outcoe, let ( i,..., i M ) be the uber of type balls that have bee reoved, ad let L ( i,..., i M ) be the last type to have bee reoved, at the first oet whe at least oe ball of each type has bee reoved. Siilarly, let ( i,..., im) be the uber of type balls that have bee reoved, ad let L ( i,..., im ) be the last type to have bee reoved, at the first oet that all the balls of ay of the types have bee reoved. It is easy to see that + ( im,..., i), if L( i,..., im) ( i,..., im) ( im,..., i), if L( i,..., im) () (,..., ) (,..., ) () L i im L im i Let ad L be the ubers of type balls reoved ad the last type reoved, respectively, i the dual proble, ad let ad L be the correspodig quatities for the origial proble. Because all outcoes are equally liely, it follows fro () ad () that (, ), P + rl if P ( rl, ) P ( rl, ), if (3)

7 The Hypergeoetric Coupo Collectio 7 I additio, it follows fro () that i im + M im i (,..., ) (,..., ) iplyig, because all outcoes are equally liely, that P ( ) P ( M+ ) (4) where ad are the ubers of balls reoved i the origial proble ad i the dual proble, respectively. Rears (a) The dual proble was previously cosidered by El-eweihi et al.(978), where their ai cocer was deteriig the probability ass fuctio of the type of the last ball reoved. I additio, they coectured that was a icreasig failure rate rado variable. This coecture was the prove by Ross et al. (98). (b) The duality result give by Equatio (4) was previously give i Ross et al. (98). REFERECES [] Adler I., Ore S., Ross S. M. (3), The coupo collector s proble revisited; Joural of Applied Probability, 4; [] El-eweihi E., Proscha F., Sethuraa J. (978), A siple odel with applicatios i structural reliability, extictio of species, ivetory depletio ad ur saplig; Advaces i Applied Probability, (); [3] Ross S. M. (), Probability odels for coputer sciece, Acadeic Press. [4] Ross S. M., Shahshahai, M., Weiss G. (98), O the uber of copoet failures i systes whose copoet lives are exchageable, Matheatics of Operatios Research, 5(3);