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A NOTE ON THE DISTRIBUTION OF THE TIME OF THE FIRST k-RECORD

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1998年度日本オペレーションズ。リサーチ学会 春季研究発表会

て −A−11

ANOTEONTHEDISTRIBUT王ONOF

TⅢETIMEOFTHEF量RSTk・RECORD

O1303783 愛知大学 玉置光司 TAMAKIMitsushi

Weexplicitlygivetheprobabilitymassfunctionandtheprobabilitygenemtingfunctionofthe firstk−TeCOrdindexfbrasequenceofimdependencamdidenticallydistributedmndomvariablesthat takeonamnitesetorpossiblevalues．Wealsocomputeits払ctorialmoments． 且。‡N甘監ODUCT五⑬NÅND＄冊ヱ岨ÅRV LetXl，X2，…beindependentandidenticallydistFibuted汽mite−Valuedrandomvariableswith probabilitymassfunctiom pi＝P（Ⅹ＝i．），i＝1，…，m． ForamxedposltlVeintegerk，k＞1，血erandomvariable

T＝min（n‥Ⅹn≦Ⅹifbrexacdykofthevaluesi，・i＝1，．．．，n）

iscalledtbe汽ー鈍かrgc（〉rd加dβ∬．

AdleTandRoss［1】deteTminedぬemrsttwomo甲entSOfTviaaTeCurSiveformulafbrits

PrObabilitygeneratlngfuncc且OmObmi皿edbyconditionlngOnarandomvariableTj，Whichisdenned

asthe鮎stk−reCOrdindexwhentheobservedraLmdomvariableshaveprobabilitymassfunction

P（Ⅹ＝i）＝ 且〒J，・・・，m・ 巧＋…＋pm’ Inぬ五s皿Ote，WedeFivechep釘Obabil五電ymassfunctionandtheprobabilicygeneratingfunctionofT expl丘c加Iya皿d也emdetermineitsねc伍Oridmoments． THEOREM 且JeCai＝pi＋・・・＋pm，i＝1，…，m．ThendlepTObabilitymassfuncciono［Tis騨Venby

二1（且一軒k，m≧k・

叩＝n｝＝霊pi（芸糎 i＝1

MoTeOVer血ep刑）bab血ygeneratimgfunc紅ionP（S）＝鴎T］isgivenby

（且） P（S）＝墓pi可 王ト1 叫S （2）

LetPj（S）＝蜂用j＝且，‥．，m．ThenAdlerandRoss軋Theoreml］givearecursive

野鼠⑬⑬厨 ねmula

岬＝賢叫1一紳減封j＝1，…副

W払五ch，COmbinedwithPm（S）＝Sk，Canbeeasilysolvedtoyield −24− © 日本オペレーションズ・リサーチ学会. 無断複写・複製・転載を禁ず.

(2)

巧（S）＝姜 lヨ k−1 叫S Thus，Sinceal＝1，Wehave m P（S）＝Pl（s）＝∑ i＝1 l r ︵ S p 叫S 彗 l ︵ ︼ l k−1 叫S denotestheprobabilitygeneratlng Whichyields（2）．（1）isimmediatefrom（2），because 1−（1一句）s

functionofthenegativebinomialrandomvariablewithparametersk−1andai・andhencecanbe

expandedinto 00 ）k￣1＝主（：二紳1（1−納n￣1・ 叫S Thustheproofiscomplete・ ThefbllowlngCOrOllaryglVeSthefactorialmomentsofT・ COROLLARY Forj＝0，1，2，‥・，

恥−1）（叫2）‥・（T−1）】菓閉脚‥樹 ・．i＝1

PROOF Fbrj＝0，1，2，…，Wehavefrorn（1）that 可（Tl」−1）（T＋j−2）…（T−1）】 00 ＝∑（n・k十j−1）（n・ktj−2）・・・（n・k−1）P（T＝n・k） n＝0 （kJj−1）（k＋j−2）…（k−1） （k＋j−1）（k＋j−2）…（k−1）

（m＋伸一げ）

1 〈 00 ∑ n＝0

m∑忠∑畠

p純k− pi中一1町（k十j），

whichisthedesiredresult．ThelastequalityfbllowsfromthewellkLlOWnidentity（See，e・g・，

Ross【2】，p・153）

）n＝a叫触・叫≧1・

00 計＋㌣￣1ト Re鮎remces

l．Adler，I．，＆Ross，S．M．（1997）．Distributionofthetimeofthefirstk−reCOrd・Probabilityin

血肋g血eer血gα〝dJゆmαfわ朋J放ね〝Ce∫11：273−278・

2．Ross，S．M．（1997）．LniTVductiontoprobabilitymOdels，6thed・NewYork：AcademicPress・

3．Tamaki，M．（1998）．Anoteonthedistributionofthetimeofthefirstk−reCOrd・Probabilityin

血助g血ee血gd〃dJゆmα血乃αJ∫cfe〝Ce∫12（toap匹ar）・ ー25− © 日本オペレーションズ・リサーチ学会. 無断複写・複製・転載を禁ず.

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