REMARKS ON THE FORMULA FOR THE MOMENTS OF THE PÓLYA PROBABILITY DISTRIBUTION

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1 ACTA UNIVERSITATIS LODZIENSIS FOLIA OECONOMICA 3(314) Tdeusz Gesteo * REMARKS ON THE FORMULA FOR THE MOMENTS OF THE PÓLYA PROBABILITY DISTRIBUTION Abstct. The pobbiity distibutio of do vibe c be chcteized by soe ubes ced petes of the distibutio. Moets beog to the ost fequety used petes. We focus o the Póy distibutio becuse we c esiy obti fo it s speci cses soe distibutios ipott i sttistics such s: bioi egtive bioi d Poisso (the st oe i the iit pocedue). I 1972 G. Mühbch gve vey iteestig foue fo the oets of the Póy distibutio. The utho did ot ivestigte the evutio of the ueic efficcy of the fou fo the oets. We wi show tht it is possibe to deostte this fou i sipe fo which is ipott fo pctic poit of view. Keywods: oets of the pobbiity distibutio Póy pobbiity distibutio. 1. INTRODUCTION I picipe do vibe is descibed ecty eough by its pobbiity distibutio. Howeve pctic esos ipose the eed to fid soe ueic chcteistics of distibutio sice they e shot desciptios d give quic copiso of the distibutios with theseves. I theoetic sttistics s we s i sttistics used i ecooy eed fequety ises to get the bsic popeties of ivestigted poputio. The we do ot etio y pticus d we chcteize the eeded popeties y tie with the hep of oe o few ubes. Fist of we use hee the ithetic e e devitio d i the cse of distibutio evidety the oets. 2. THE G. PÓLYA DISTRIBUTION I the ppe we study fou fo the oets of the G. Póy pobbiity distibutio give i 1972 by G. Mühbch. Fo this pupose we shoud be i id tht this distibutio is epessed by the fou * Pofesso Eeitus Uivesity of Łódź Pod e-i: tdge@th.ui.odz.p [9]

2 10 Tdeusz Gesteo p( p)...[ p( 1) ] q( q)...[ q( 1) ]) PX ( ) 1(1 )(1 2 )...[1 ( 1) ] whee 0 p y ube but fo 0 we ssue i( p q) q 1 p. To fciitte the ottio of tht epsibe fou we usuy e use of the so ced fctoi poyois of the degee i espect of (so ow s geeized powe of the degee of the ube ) i the foowig wy 0 [ ] [ 1 ] whee =12 deotes y ube It foows fo the give ecuece defiitio tht 2 1 Bsig o these foue we c deote the Póy distibutio s foows p PX. q 1 G. Mühbch deoted tht distibutio with sighty diffeet sybos q whee we c eebe tht G. MÜHLSBACH FORMULA FOR THE MOMENTS To fid fou fo the oets of the Póy pobbiity distibutio G. Mühbch used opeto Q [ ; ] which is peseted s foows f Qf; f q 0

3 Res o the Fou fo the Moets of the Póy Pobbiity Distibutio 11 whee f deotes fiite diffeece of degee of fuctio f ( ) with defied s foows [01] f f 0 whie f ( ) f f 1 1 = 012 q P s give peviousy. O the bsis of tht opeto the utho hs obtied the foowig fou fo the oets whee Qg; g t. q 0 g t t t wht we c so deote i the fo o i bette ow sybos g t 1 0 whee t this ottio used = p. 0 ( 1 g t ) 4. A MODIFICATION OF THE MÜHLBACH FORMULA The utho did ot get isight ito the thetic efficcy of the give fou. Tht fou c be peseted i sipe oe hdy d esie fo by use of the Stiig ubes of the secod id which e defied

4 12 Tdeusz Gesteo (see: J.Łusiewicz d M. Wus (1956) p. 50) s the coefficiets t the fctoi poyois i the idetity S S [ ] S S S whee [ ] ( 1)( 2)...( ( 1)) d gettig S 1 S 0 fo S 1 fo S 0 fo d usig the fiite diffeece of zeo i.e. the diffeece of the fuctio t the poit = 0 with the step 1 tht is y d so o. We ow the foowig fou which is used fo pepig the tbes of the fiite diffeece of zeo Tig ito ccout ! d the fct tht the diffeece of the degee i the fou fo the oets is ccuted t the poit zeo we obti the ottio of the fou i the fo 0. 0! 1 The etio betwee the fiite diffeeces of zeo d Stiig ubes of the secod id is s foows:

5 Res o the Fou fo the Moets of the Póy Pobbiity Distibutio 13 0! S Theefoe the fou fo the oets c be witte i the fi fo. S 0 1 The Stiig ubes e tbuted fo epe t A. Kuf (1968) p. 52 wht ows to ccute the oet of the eeded degee efficiety eough. Fo epe S ( 1)( 2) ( ) 2 4 ( ) ( )( 2) 7( 1) 6( 1)( 2) 1 (1 )(1 2) ( )( 2)( 3) ( 1)( 2)( 3) (1 )(1 2)(1 3) 5. A RECURRENCE RELATION FOR THE MOMENTS I the cdeic boo of the pobbiity theoy by T. Gesteo d T. Śód (1972) ecuece etio fo the oets (fou (6.5.11) p. 227)is give togethe with poof i the Póy do u schee tht is whe N is ube of bs i the u b is ube of white bs i the u c is ube of bc bs i the u b + c = N s is ube of the puttig o tig out of the bs of give coou ccodig to the coou of the b dw by the d tued to the u. The fou is of the fo 1 1 b bs s N s i0 i i1 i2 i

6 14 Tdeusz Gesteo whee = 012 is the ube of eized epeieces (dws). I the cse s<0 it shoud be ssued: s b d sc I the Póy schee we s bout the pobbiity of obtiig white bs i dws. If we egd the ow i tht schee etio: b p c q s we obti cofotbe fo of the fou fo the N N N oets of the Póy distibutio 1 1 p p 1 i0 i i1 i2. i Fo tht fou fo the oets of the Póy distibutio oe c esiy obti s speci cses the foue fo the oets of the distibutios: bioi (Beoui) hypegeoetic egtive bioi d i the iit cse so fo the Poisso distibutio. REFERENCES Gesteo T. Śód T. (1972) Kobitoy i chue pwdopodobieństw PWN Wszw. Kuf A. (1968) Itoductio à Cobitoique e Vue des Appictios Pis Duod. Łusiewicz J. Wus M. (1956) Metody ueycze i gficze Część I Wszw PWN. Mühbch G. (1972) Reusiosfoe fü die zete Moete de Póy- ud de Bet- Veteiug Meti 19 Fsc Tdeusz Gesteo UWAGI O WZORZE NA MOMENTY ROZKŁADU PRAWDOPODOBIEŃSTWA G. PÓLYI Steszczeie. Rozłd pwdopodobieństw zieej osowej oże być schteyzowy pzez podie pewych iczb zwych peti ozłdu. Do jczęściej używych petów eżą oety. Uwg sz jest socetow ozłdzie pwdopodobieństw G. Póyi bowie oż z iego łtwo uzysć jo pzypdi szczegóe ub odpowiedio gicze wże w sttystyce ozłdy j dwuiowy ujey dwuiowy ub Poisso. W G. Mühbch podł iteesujące wzoy oety ozłdu Póyi. Auto te ie wił w oceę efetywości chuowej podego wzou oety zwyłe. Pożey co zczeie ptycze że wzó te oż pzedstwić w postszej wygodej foie. Słow uczowe: ozłd pwdopodobieństw Póyi oety ozłdu.