The Coupon Collector Problem in Statistical Quality Control

Transcription

1 The Coupo Collector Proble i Statitical Quality Cotrol Taar Gadrich, ad Rachel Ravid Abtract I the paper, the author have exteded the claical coupo collector proble to the cae of group drawig with iditiguihable ite The reult are applied to a tatitical quality cotrol proble that arie i a dairy' bottle fillig proce with ozzle A equetial ized aplig i ade i order to detect the ozzle The preet reearch cocer the uber of aplig, called the waitig tie, required util each ozzle i detected at leat oce Boe Eitei tatitic i ued to aalyze the waitig tie ditributio ad a uerical exaple i give Idex Ter Boe Eitei tatitic, iditiguihable ite, waitig tie I INTRODUCTION The claical coupo collector proble (CCCP) cocer a hopper who trie, i everal attept, to collect a coplete et of differet coupo Each attept provide the collector with oe coupo radoly choe fro ow type, ad there i a uliited upply of coupo of each type Referece [] aalyzed the CCCP a a occupacy proble He derived a expreio for the expected waitig tie, ie the uber of attept eeded to obtai the coplete et Extedig CCCP ha bee a challege for reearcher durig the lat few decade May of the exteio that have bee developed have bee foud to be ueful odel for cietific ad egieerig applicatio Referece [2] derived etiator for the ea ad variace of the waitig tie for the uequal probability cae Their reult ca be ued to aalyze the ea tie util a rado wal o a tar graph viit ditict leaf ad the retur to the origi The IP tracebac proble i coidered i [3] He derived boud for the copleetary cuulative ditributio fuctio of the detectig cot (waitig tie) Thee boud are very helpful for evaluatig the efficiecy of variou PPM chee Several reearche exteded the CCCP to the group drawig cae, ie each attept provide the collector with a group of ditict coupo A exact olutio for the ditributio ad factorial oet related to Maucript received February 27, 2008 T Gadrich i with the Idutrial ad Maageet Departet, ORT BRAUDE College of Kariel, Irael (phoe: ; fax: ; e-ail: taarg@braudeacil) R Ravid i with the Idutrial ad Maageet Departet, ORT BRAUDE College of Kariel, Irael (phoe: ; fax: ; e-ail: rachelr@braudeacil) the waitig tie for obtaiig a pecified ubet of coupo i reported by [4] Referece [5] exteded the reult i [4] to rado group ize with uequal probabilitie They deteried the expected waitig tie ad gave boud o thi uber A applicatio to reliability egieerig wa alo give CCCP with equal probabilitie ad rado ize aple are reearched i [6] They coputed the ditributio ad ea for the uber of aplig eeded to obtai j coupo type, give that there are curretly i coupo type I the preet wor, CCCP i geeralized to the group aplig cae i which the aple ite are ot ecearily ditict The waitig tie ditributio ad it factorial oet are coputed i Sectio 2 I Sectio 3 the reult are applied to a tatitical quality cotrol proble that arie i a dairy' bottle fillig proce ad a uerical exaple i give II WAITING TIME DISTRIBUTION Coider a populatio that coit of type of ite, each of which ha a uliited uber of copie Sequetial ized aplig are ade, whe the ite i each aple are ot ecearily ditict, util ay type of a coplete et (ie, all type of ite) i obtaied at leat oce I the particular cae that =, we have a CCCP I order to derive explicit forula for the probability ditributio fuctio ad probability geeratig fuctio of the waitig tie, we eed oe preliiary reult cocerig the uber of differet ite type achieved after aplig We begi with the followig lea, Lea : Let B j j,2,, deote the evet that a type j ite wa ot detected after aplig We have 2,2, PBj () j,2,, Proof: A igle ized aple, choe fro a type populatio, ca be decribed i ter of a rado ditributio of iditiguihable ball ito cell Uig Boe Eitei tatitic [] we get that it ca be doe i 2 equiprobable arrageet There are arrageet i which oe cell i epty Sice ucceive aplig are tatitically idepedet, the reult follow

2 Defie the rado variable X to be the uber of differet ite type achieved after aplig; Theore : The ditributio of X i give for every (=,2,,) by: 0 P X P X (2) (3) 0 Proof: Defie A j,, ( ) j to be a fixed ized ubet of the coplete type et The probability that thee ite type will be obtaied after aplig equal: j j P B P B Uig the icluio-excluio forula, the probability that at leat oe of the ite type i A () will ot be obtaied i aplig i: P Bj Obviouly, 0 P X Equatio (3) i derived fro (2) with the aid of Lea 2 i [4] The cuulative ditributio fuctio i P X P X i i i i i i i 0 i i 0 iax, i i 0 iax, i 0, which derived (3) Corollary : The expected uber of ditict ite type that are achieved after a erie of aplig i give by 2 E X, 2, Proof: For j defie X 0 I j ite j ha ot bee obtaied durig aplig otherwie Oe ca eaily ee that: I j Clearly, j 2 PI j E I j PBj Uig the additive property of expectatio, we get: 2 j j E X E I We ow retur to our ai purpoe: deteriig the waitig tie Let Z be the uber of aplig eeded util oe collect at leat ite type fro the coplete et (4)

3 Theore 2: The ditributio of (for every ) by: P Z Z i give (5) 0 Moreover, the probability geeratig fuctio (pgf) of Z i give by G Z (6) 0 Proof: The probability ditributio fuctio of Z ca be obtaied fro (3) uig the followig relatio: P Z P X P X G The pgf of Z Z E Z i defied a: 0 (7) Equatio (6) follow (7) i the cae that i le tha Corollary 2: The p-factorial oet p N of Z are give by: E Z Z Z p p! 0 p p Proof: The reult i (8) follow (7) by derivig the pgf of Z accordig to, p tie ad ubtitutig Corollary 3: The expectatio ad the variace of the waitig tie util we achieve the coplete et are give by: E Z 0 E Z (8) (9) (0) 2! 0 Z 2 VAR Z E Z Z E Z E Z () Proof: Forula (9) ad (0) are derived fro (8) by ubtitutig p= ad p=2, repectively; ad = III NUMERICAL EXAMPLE The odel decribed i Sectio II ha a direct applicatio to a tatitical quality cotrol proble Coider a dairy' bottle fillig proce coitig of a 24-ozzle achie After the bottle are filled with il, they are gathered i a collectio area Each hour a rado aple of five bottle i draw fro the collectio area ad teted i order to cotrol the fillig proce There i o arig o a bottle idicatig which ozzle filled it The quality egieer wat to ow, after how ay hour o the average, bottle filled by ay oe of the ozzle will be teted Uig the otatio of Sectio II, the 24 ozzle for a 2

4 Expectatio Z coplete et of type (=24) A hourly rado drawig of five bottle defie the aple ize =5 Sice the bottle are ot ared, the odel that aue that the ite i each aple are ot ecearily ditict i uitable here Uig (2), the probability ditributio fuctio of Z 24 i how i Fig a a fuctio of the uber of hourly drawig that are doe i order to tet all the ozzle Tie (hour) Fig Waitig tie probability ditributio fuctio for =,2,,50 (=24, =5) Uig (9), the expectatio of Z 24 i how i Fig 2 a a fuctio of the aple ize Thi ca be ueful for the quality cotrol egieer If there i ay cotrait o how ay hourly aplig he ca draw, the aalyt will eaily fid the required aple ize for detectig the coplete et Saple Size Fig 2 Waitig tie expectatio a a fuctio of the aple ize (=24) I Table, for =24, the expectatio ad tadard deviatio of the waitig tie a a fuctio of the aple ize (ie, a a fuctio of ) are give for the followig two cae: I each aplig a et of differet ite i draw The factorial oet related to the waitig, i thi cae, were obtaied i [4] by: p E Z Z Z p j p! j0 j j (2) j j p The expectatio ad the tadard deviatio follow fro (2) 2 I each aplig, a group of ite i draw, but the ite are ot ecearily ditict The expectatio ad the tadard deviatio ca be coputed uig (9) ad (), repectively Subtitutig =, i (8)-(2) yield the ow reult for the CCCP whe =24 O average, oe eed aroud 9 aplig i order to be ure of detectig all 24 differet type I the diary proble whe =5; the calculatio yield that o average, after 20 hour all the ozzle will have bee exaied Table Expectatio ad tadard deviatio of the waitig tie =24 Ditict ite Iditiguihable ite Drawi g group Expecta- SD Expecta- SD ize tio tio IV CONCLUSIONS A well ow proble, called the CCCP, i exteded to the cae of group aplig with iditiguihable ite, uig a

5 occupacy odel Clearly, group drawig help to reduce the expected uber of aplig required i order to detect the coplete et The probability geeratig fuctio ha bee developed for coputig the expectatio ad tadard deviatio of the waitig tie Quality egieer ay fid the give expreio ueful for cotrollig procee uch a the bottle fillig proce decribed i thi paper REFERENCES [] W Feller, A Itroductio to Probability Theory ad It Applicatio, Wiley: New Yor, Third Editio, 970, ch 4 [2] E Peoz ad S M Ro, Applied Probability ad Stochatic Procee, vol 9, J S Shathiuar ad U Suita, Ed Boto: Kluwer, 999, pp [3] S Shioda, Soe upper ad lower boud o the coupo collector proble, Joural of Coputatioal ad Applied Matheatic, vol 200, 2007, pp 54 67, [4] W Stadje, The collector proble with group drawig, Advace i Applied Probability, vol 22, 990, pp [5] I Adler ad S M Ro, The coupo ubet collectio proble, Joural of Applied Probability, vol 38, 200, pp [6] J E Kobza, S H Jacobo, ad D E Vaugha, A urvey of the coupo collector proble with rado aple ize, Methodology ad Coputig i Applied Probability, vol 9(4), 2007, pp