An Alternative Approach of Binomial and Multinomial Distributions

Transcription

1 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) A Alterative Approach of Biomial ad Multiomial Distributios Mia Arif Shams Ada Departmet of Mathematical Sciece, Ball State Uiversity, Mucie, IN 47304, USA maada@bsu.edu Md. Tareq Ferdous Kha Departmet of Statistics, Jahagiragar Uiversity, Dhaa-34, Bagladesh Md. Forhad Hossai Departmet of Statistics, Jahagiragar Uiversity, Dhaa-34, Bagladesh Abdullah Albalawi Departmet of Mathematical Sciece, Ball State Uiversity, Mucie, IN 47304, USA Received 4 April 05 Accepted 8 September 06 Abstract I this paper we have tried to preset a alterative approach for two discrete distributios such as Biomial ad Multiomial with a ew cocept of samplig havig a more geeral form apart from the traditioal methods of samplig. The eistig distributios may be obtaied as a special case of our ewly suggested techique. The basic distributioal properties of the proposed distributios are also bee eamied icludig the limitig form of the proposed Biomial distributio to ormal distributio. Keywords: Geeralized Biomial Distributio, Geeralized Multiomial Distributio, Samplig Methods, Distributioal Properties, Arithmetic Progressio, Limitig Form. Mathematics Subject Classificatio: 60E05, 6E0, 6E5, 6E0.. Itroductio The discrete distributios are widely used i the diversified field ad amog them the distributio lie Biomial, Multiomial, Poisso ad Geometric are the most commoly used discrete distributios. The other discrete distributios iclude uiform or rectagular, hypergeometric, egative biomial, power series etc. The trucated ad cesored forms of the differet discrete distributios are used as probability distributio i statistics literature ad i real life problems. The biomial distributio was first studied i coectio with the games of pure chace but it is ot limited withi that area oly, where the Multiomial distributio is cosidered as the geeralizatio of the biomial distributio. The umber of mutually eclusive outcomes from a sigle trial are i multiomial distributio compared to two outcomes amely success or failure of Biomial distributio. The usual biomial distributio is the discrete probability distributio of the umber of successes 0 to resulted from idepedet Beroulli trails each of which yields success with probability p ad failure with Copyright 07, the Authors. Published by Atlatis Press. This is a ope access article uder the CC BY-NC licese ( 69

2 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) probability q. Oe of the importat assumptios regardig biomial variate represetig the umber of success is that it ca tae oly values i the sequece of 0,,,,. But i real world, the successes of biomial may ot occur i the usual way rather it may occur i a differet sequece such as (i) 0,,4,,, (ii),4,,, (iii) 0,3,6,3, (iv) 3,6,9,,3 ad so o. I cases (ii) ad (iv), trucated distributio is the better optio to fid the probability of umber of success. I trucated distributio, it is assumed that the trucated values of the radom variable have certai probabilities. If it is cosidered that there is o eistece of the trucated values, the trucated distributio caot provide the probability due to mathematical cumbersome. I the remaiig cases, Biomial ad trucated Biomial are completely helpless. I this cotet, we have suggested a alterative approach of Biomial ad Multiomial distributio havig geeralized sequece of the values of the radom variables. For coveiece the distributios are defied as relatively geeralized distributio. The umber of successes of the proposed distributios is represeted by the arithmetic progressio, where, a is o-egative iteger ad termed as miimum umber of success, d is positive iteger represetig the cocetratio of success ad is a o-egative iteger idicatig the total umber of trails. To justify the sequece with real life situatio, let us cosider a eample of umber of defective shoes. It is well ow that the shoes are produced pair wise. That is, if we mae attempts to idetify the umber of defective shoes, the it is usual that the umber of defective shoes would occur pair wise. I this cotet, the umber success i the form 0,,4,, is justified. If it is ow that the miimum umber of defective items i attempts are a, the these occurreces are ot chace outcome ad may be regarded as costat ad the we are iterested to fid the probability of chace outcome taig form a, a +, a + 4,,, where a > 0 ad is o-egative iteger. That is, umber of defective shoes larger tha a. The other possible sequeces are also justifiable i this way by real life eamples. Oe may cofuse the proposed form of the distributio with the usual trucated distributio. The major differece is that i our form the probability eists oly for the possible umber of success show i the sequece. I the eample discussed above, the defective umber of shoes tae the values 0,,4,,. This idicates that there is o eistece of the umber of defective item(s) of the form,3,5, ad thus o probabilities. O the other had, i trucated distributio, there is eistece of the umber of success which is trucated ad also they have the probabilities. A umber of authors published their wor uder the headig of geeralized biomial distributio but they were differet i terms of the ey cocept of our preset wor. Altham showed two geeralizatios of the biomial distributio whe the radom variables are idetically distributed but ot idepedet ad assumed to have symmetric joit distributio with o secod or higher order iteractios. Two geeralizatios are obtaied depedig o whether the iteractio for discrete variables is multiplicative or additive. The distributio has a ew parameter θ > 0 which cotrols the shape of the distributio ad fleible to allow for both over or uder-disperse tha traditioal Biomial distributio. Whereas, the beta-biomial distributio allows oly for over-disperse distributio tha the correspodig Biomial distributio (Johso, Kemp ad Kotz ). Dwass 3 have provided a uified approach to a family of discrete distributios that icludes the hypergeometric, Biomial, ad Polya distributios by cosiderig the simple sample scheme where after each drawig there is a replacemet whose magitude is a fied real umber. Paul 4 derived a ew three parameter distributio, a geeralizatio of the biomial, the beta-biomial ad correlated beta-biomial distributio. Further a modificatio o beta-correlated biomial distributio was proposed by Raul 5. I the geeralizatio of the probability distributio, Paaretos ad Xealai 6 developed cluster biomial ad multiomial model ad their probability distributios. I his study, Madse 7 discussed that i may cases biomial distributio fails to apply because of more variability i the data tha that ca be eplaied by the distributio. He poited out a characterizatio of sequeces of echageable Beroulli radom variables which ca be used to develop models which is more fluio tha the traditioal biomial distributio. His study ehibited sufficiet coditios which will yield such models ad show how eistig models ca be combied to geerate further models. 70

3 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) A geeralizatio of the biomial distributio is itroduced by Drezer ad Farum 8 that allow the depedece betwee trials, o-costat probabilities of success from trial to trial ad which cotais usual biomial distributio as special case. A ew departure i the geeralizatio was carried out by Fu ad Sproule 9 by adoptig the assumptio that the uderlyig Beroulli trials tae o the values α or β where α < β, rather tha the covetioal values 0 or. This redered a four parameter biomial distributio of the form B(, p, α, β). I a recet wor, Altham ad Hai 0 itroduced two geeralizatios of multiomial ad Biomial distributios which arise from the multiplicative Biomial distributio of Altham. The forms of the geeralized distributios are of epoetial family form ad termed as multivariate multiomial distributio ad multivariate multiplicative biomial distributio. Lie the Altham s geeralized distributio, both the distributio has a additioal shape parameter θ which correspods usual distributio if it taes value ad over ad uder-disperse for greater ad less tha respectively.. New Approach of Biomial Distributio I the traditioal Biomial samplig each time we draw a sample havig either a success or a failure, we cotiue this method up to trials ad we may have umber of successes startig from 0 ad may eded up to, but i real world, the success of Biomial distributio may ot occur i the usual sequeces rather it may happed as i) 0,,4,, ii),4,, iii) 0,3,6,3 iv) 3,6,9,,3 ad so o. These idicate that the umber of success may follow a arithmetic progressio, where, a is a o-egative iteger represetig the miimum umber of success, d is a positive iteger represetig the cocetratio of success occurrig ad is a o-egative iteger idicatig the total umber of trails. Defiitio.. A radom variable X is said to have a relatively geeral biomial distributio if it has the followig probability mass fuctio p q P(; a,, d, p) a p q + a p q ; a, a + d, a + d,, (.) where, a 0 is the miimum umber of success, d > 0 is the cocetratio of occurrece of success, is a predefied fiite umber of o-egative iteger represets the umber of trials ad p is the probability of success such that p + q ad a p q is a costat. The probability mass fuctio P(; a,, d, p) stads for probability of gettig success out of maimum of successes i trials. Theorem.. Show that the probability mass fuctio of geeralized biomial distributio is a probability fuctio. Theorem.. Show the relatioship betwee geeralized biomial distributio with parameter a 0, d > 0, 0 ad p ad usual biomial distributio with parameter 0 ad p. 7

4 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) It ca be easily show that the probability mass fuctio of geeralized biomial distributio reduces to the probability mass fuctio usual biomial distributio whe a 0. Theorem.3 Show that the momet geeratig fuctio of geeralized biomial distributio is a M X (t) a (pe t ) q p q (.) Theorem.4. The mea ad variace of geeralized biomial distributio are ()p ad ()pp respectively. Theorem.5. If X follows the geeralized biomial distributio the fid the 3 rd ad 4 th raw ad cetral momets of X respectively are μ 3 () 3 p 3 3() p 3 + ()p 3 + 3() p 3()p + ()p (.4) μ 4 () 4 p 4 6() 3 p 4 + () p 4 6()p 4 + 6() 3 p 3 8() p 3 + ()p 3 + 7() p 7()p + ()p (.5) μ 3 ()pq( p) (.6) μ 4 ()pp + 3 () pp (.7) Still the special case for 3 rd ad 4 th raw ad cetral momets holds for a 0 ad d ad tured to the form of usual Biomial distributios. Theorem.6. The measures ad coefficiet of sewess ad urtosis of geeralized biomial distributio are: Sewess Measures of Sewess (β ) ( p) ()pq (.8) p Coefficiet of Sewess (γ ) β ()pq (.9) From coefficiet of sewess, the followig coclusio ca be draw ad still surprisig that the ature of sew depeds oly o p oly ad which is similar to the usual biomial distributio as: i) The distributio is positively sewed if p <. ii) O the other had, the distributio is egatively sewed if p >. iii) Ad the distributio is symmetric if p. Kurtosis ( 6pp) Measures of urtosis (β ) 3 + ()pp (.0) 7

5 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) ( 6pp) Coefficiet of urtosis (γ ) β 3 ()pp (.) These equatios tell us that the geeralized distributio is i) Mesourtic if. 6 ii) Platyurtic pp > ad 6 iii) Leptourtic if pp <. 6 Theorem.7 Show that the maimum lielihood estimate of the parameter p is umber of success from maimum of success i trials. Theorem.8. The MLE estimator of umber of trial of the distributio is d i i p, where is the total a. Theorem.9. Necessary coditios to derive the geeralized Poisso distributio from geeralized biomial distributio. Geeralized Poisso distributio ca be derived from the geeralized biomial distributio uder the followig assumptios: i) p, the probability of success i a Beroulli trail is very small. i.e. p 0. ii), the umber of trails is very large. i.e.. iii) ()p λ is fiite costat, that is average umber of success is fiite. Uder this coditio, we have ()p λ p λ () ad q λ () Theorem.0 Normal distributio as a limitig form of geeralized biomial distributio. 3. New Approach of Multiomial Distributio Uder the samplig scheme described i Sectio, cosider the situatio where oe of the mutually eclusive outcomes is possible from a sigle trial other tha oly success or failure. More specifically, if the outcomes are deoted by e, e, e,, e ad the umber of occurreces of the respective outcomes are i deoted by,,, such that i, where a is o-egative iteger ad termed as miimum umber of success, d is positive iteger represetig the cocetratio of success ad is a oegative iteger idicatig the total umber of trails, the our suggested geeral form of Biomial as well as traditioal Multiomial distributio caot provide the probability that the evet e occurred times, the evet e occurred times ad so o the evet e occurred times. To overcome this situatio, we have suggested the ew approach of Multiomial distributio ad termed as relatively more geeral Multiomial or geeralized Multiomial distributio. I this sectio we would preset oly the defiitio of the suggested distributio ad statemet of the theorems that we have derived for our preset wor to eep the paper size stadard ad other reaso is that the derivatios are very much similar to the suggested Biomial distributio. Defiitio 3.. discrete radom variable X, X,, X is said to have a geeralized multiomial distributio if it has the followig probability mass fuctio 73

6 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) P(,,, ; a,, d, p, p,, p )!!! p p p!!! p p p ()!!!! p p p ()!!!! p p p,,, a,,, a (3.) where, a 0, d > 0, 0 ad p, p,, p such that ad i i. i p i, are the parameters of the distributio. The defied fuctio is a probability mass fuctio as it holds the coditios of the probability fuctio ad easily reduces to traditioal Multiomial distributio if a 0 ad d ad thus may be said that the traditioal Multiomial distributio is a special case of the suggested Multiomial distributio. Theorem 3.. Geeralized Biomial distributio is a special case of geeralized Multiomial distributio. Theorem 3.. The momet geeratig fuctio of geeralized Multiomial distributio is M X,X,,X (t, t,, t ),,, a,,, a ()! (p i i! i e t i) i i ()! p i! i i i i (3.3) ad the mea ad the variace of X i (i,,, ) respectively are E(X i ) μ i ()p i (3.4) V(X i ) ()p i ( p i ) (3.5) Theorem 3.3. The 3 rd ad 4 th raw ad cetral momets of geeralized Multiomial distributio are μ 3i (a + d) 3 p 3 i 3() p 3 i + ()p 3 i + 3() p i 3()p i + (a + )p i (3.6) () 4 p 4 i 6() 3 p 4 i + () p 4 i 6()p 4 i + 6() 3 3 p i μ 4i 8() p 3 i + ()p 3 i + 7(a + d) p i 7()p i + ()p i (3.7) μ 3i ()p i ( p i )( p i ) (3.8) ad μ 4 ()p i ( p i ) + 3 () p i ( p i ) (3.9) I all the momets if we put a 0 ad d the these reduces to the form of momets of traditioal Multiomial distributio which agai justifies that the traditioal Multiomial distributio is a special case of our suggested geeralized Multiomial distributio. 74

7 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) Theorem 3.4. Karl Pearso s measures ad coefficiets of sewess ad urtosis of geeralized multiomial distributio. Measure of Sewess: ( p i ) β i ()p i ( p i ) Coefficiet of Sewess: p i γ i β i ()p i ( p i ) Measure of Kurtosis: β i 3 + { 6p i( p i )} ()p i ( p i ) (3.0) (3.) (3.) Coefficiet of Kurtosis γ i β i 3 { 6p i( p i )} ()p i ( p i ) (3.3) The coclusio from the measures ad coefficiet of sewess ad urtosis of the geeralized Multiomial distributio ca be draw i the same way as draw i the geeralized Biomial distributio depedig o the values of p i. Theorem 3.5. The maimum lielihood estimator of the parameters of geeralized multiomial distributio is p ı i, where () i is the umber of success comprisig ( i ) is the total umber of failure subject to coditio that maimum umber of success is. 4. Discussio ad Coclusio We have suggested relatively more geeral form of two discrete distributios such as Biomial ad Multiomial for the differet samplig scheme which is described above ad termed as geeralized distributio. It is evidet from the geeralized distributio that if samplig is draw i the usual maer, the our suggested distributios reduces to the traditioal form ad thus it may coclude that the traditioal Biomial ad Multiomial distributio are the special cases of our proposed geeralized Biomial ad Multiomial distributio. Lie the traditioal distributios, all of the distributioal properties icludig limitig theorems of the suggested distributios have derived. The trucated cases of the traditioal distributio ca be address more accurately by our ew approach of the distributios. I geeral, the ew approach of the distributios are providig more access ad broade the scope from the theoretical poit of view as well as from the stadpoit of real world problem solvig. Geeralized sequece of the umber of success of the proposed distributios may be cosidered as a added advatage i the distributio theory. Refereces [] P. M. E. Altham, Two Geeralizatios of the Biomial Distributio, J. R. Stat. Soc. Ser. C. Appl. Stat. 7 () [] N. L. Johso, A. W. Kemp ad S. Kotz, Uivariate Discrete Distributios. 3rd ed. (Joh Wiley & Sos, New Jersey, 005). [3] M. Dwass, A Geeralized Biomial Distributio, Amer. Statist. 33()

8 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) [4] S. R. Paul, A Three-Parameter Geeralizatio of the Biomial Distributio, Comm. Statist. Theory Methods 4(6) [5] S. R. Raul, O the beta-correlated biomial (bcb) distributio-a three parameter geeralizatio of the biomial distributio, Comm. Statist. Theory ad Methods 6(5) [6] J. Paaretos. ad E. Xealai, O Geeralized Biomial ad Multiomial Distributios ad Their Relatio to Geeralized Poisso Distributio, A. Ist. Statist. Math. 38() [7] R. W. Madse, Geeralized biomial distributios, Comm. Statist. Theory ad Methods () [8] Z. Drezer ad N. Farum, A geeralized biomial distributio, Comm. Statist. Theory ad Methods () [9] J. Fu ad R. Sproule, A geeralizatio of the biomial distributio, Comm. Statist. Theory ad Methods 4(0) [0] P. M. E. Altham ad R. K. S. Hai, Multivariate Geeralizatios of the Multiplicative Biomial Distributio: Itroducig the MM Pacage, J. Statist. Software. 46() 0-3. Appedi Proof of Theorem.. We ow, the probability mass fuctio of the suggested geeralized biomial distributio is P(; a,, d, p) p q p q a ; a, a + d, a + d,, (.) It is clear from the fuctio that P(; a,, d, p) 0 for all values of X with differet values of a, d, ad p. Agai, P(; a,, d, p) a a a p q a p q p q p q a As P(; a,, d, p) 0 ad probability fuctio. a P(; a,, d, p), so we may coclude that P(; a,, d, p) is a Proof of Theorem.3. Accordig to the defiitio of momet geeratig fuctio, M X (t) E(e tt ) e tt P(; a,, d, p) a a > M X (t) e tt + a p q 76

9 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) > M X (t) + a (pe t ) q M X (t) a a a (pe t ) q p q (.) This is the momet geeratig fuctio of geeralized biomial distributio. Proof of Theorem.4. The mea of the distributio ca be calculated from its momet geeratig fuctio by differetiatig it with respect to t ad equatig t 0. That is E(X) μ d dd [M X(t)] t0 a (pe t ) (pe t )q a p q t0 a (pe t ) q a p q t0 a p q a p q ()( )! a ( )! ( )! pp q ( ) a p q ( )! ()p a ( )! ( )! p q ( ) a p q ()p a p q ( ) a p q ()p a p q a p q ()p E(X) μ ()p (.3) which is the mea of the distributio. Similarly, it ca be foud that the variace of the distributio is (a + d)pp. If we substitute a 0 ad d i mea ad variace of the geeralized distributio the it reduces to mea ad variace of usual distributio which holds the property that usual distributio is special case of the geeralized distributio. 77

10 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) Proof of Theorem.7. We ow, the probability mass fuctio of biomial distributio itself is a lielihood fuctio. Therefore, the lie fuctio of geeralized biomial distributio is L p q p q a + a p q (.) where, is the total umber of success from maimum of success i trials. Taig logarithm i both sides, we have lll ll + a p q > lll lll + ll + + ( )ll( p) (.3) Differetiatig equatio (.3) with respect to p ad equatig to zero, we get δ δδ (lll) + ( ) 0 p p > 0 p p p ()p p > 0 p ( p ) > ()p 0 > ()p > p Hece, the maimum lielihood estimate of the parameter p is success from maimum of success i trials. (.4), where is the total umber of Alterative Proof. The lielihood fuctio of geeralized biomial distributio ca be writte as a + d L p q a+d d p q d d p 3q d 3 d p 3 q d > L p i i q i i > L C. p i a + d d d d 3 i ( p) i i (.5) a + d where, C d d d is a costat. 3 Taig logarithm i both sides, we have lll ll C. p i i( p) i i 78

11 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) lll + i lll + i ll( p) (.6) i i Differetiatig equatio (.6) with respect to p ad equatig to zero, we get i i δ δδ (lll) p + ( i p i ) ( ) 0 > i i p i i ()p + p i i 0 p ( p ) > i ()p 0 i > ()p i > p i i i (.7) where, i i is the total umber of success out of maimum of success i trials. Proof of Theorem.8. If the probability of success p is ow from prior owledge or estimated o the basis of observatio the we ca also estimate the umber of trial. Let us defie p is the probability of success which is either ow or estimated. Therefore, p i i a + d > a + d i i p > d i i a p > d i i a (.8) p Agai, the value of a is the iitial umber of success or i other words, the umber success i first trial. Whereas, d is the differece betwee the occurred umber of success i successive trials. Proof of Theorem.0. Normal distributio ca be derived from the geeralized biomial distributio uder the followig coditios: i) The probability of success p or the probability of failure q are ot so small ad ii), the umber of trails is very large i.e. teds to ifiity. We ow, the probability mass fuctio of geeralized biomial variate X with parameter a,, d, ad p is P(; a,, d, p) p q p q a ; a, a + d, a + d,, 79

12 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) ()!! ( )! p q ()!! ( )! p q a (.9) Now, let us cosider the stadard ormal variate as E(X) z var(x) ()p ()pp (.0) Whe a, z a ()p ()pp ()p q ad whe, z ()p ()pp + a ()pp ()q ()pp ()q p Thus, i the limit as Z taes the values to Hece the distributio of Z will be a cotiuous distributio over the rage to with mea zero ad variace uity. Now, uder coditios (i) ad (ii), we shall fid the limitig form of the (.) by applyig the Stirlig s approimatio for factorials. For large, equatio (.) ca be writte as lim P(; a,, d, p) ()!! ( )! p q lim ()!! ( )! p q a (.) Let us first wor with umerator oly ()! lim! ( )! p q π e () () + lim p q π e + π e ( ) ( ) + () + lim p q π + ( ) + {()p} + {()q} + lim π ()pp + ( ) + lim π ()pp + + )p + ()q (a ( ) (.) We have from equatio (.0) ()p + z ()pp 80

13 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) > ()p + z q ()p ad ()p z p ()q Agai, dd ()pp dd From equatio (.) we have ()! lim! ( )! p q lim π ()pp + z lim + z lim lim where, N π q ()p q ()p q N + z ()p + + )p + ()q (a ( ) ()p+z ()pp+ ()p+z ()pp+ ()p+z ()pp+ p z ()q π p z ()q π p z ()q > log N ()p + z ()pp + log + z q ()p + ()q z ()pp + log z p ()q ()p z ()pp+ ()q z ()pp+ ()q z ()pp+ (.3) 8

14 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) > log N ()p + z ()pp + log z q ()p z q ()p + z3 3 3 q ()p + + ()q Z ()pp + log z p ()q z p ()q z 3 3 p ()q + > log N z ()pp z q + z q + z q ()p z 4 + z ()pp z p + z p z p ()q z 4 z > log N () q p p z + q z 4() q p p q + q ()p + p ()q + Hece, as limit, we get log N z > N ez. Now, substitutig the value of N i equatio (.5) we have ()! lim! ( )! p q z π e Puttig these value i equatio (.3) we have lim P(; a,, d, p) f(z) π e π e z z z π e ; < z < which is the probability desity fuctio of stadard ormal variate Z. If we cosider the trasformatio z ()p the the probability desity fuctio of X will be ()pp f() π ()pp e ()p ()pp ; < < (.4) If we deote the mea of X by μ ad variace of X by σ the the probability desity fuctio become f() σ π e μ σ ; < < 8

15 Joural of Statistical Theory ad Applicatios, Vol. 6, No. (Jue 07) Hece, the proof of the theorem. Proof of Theorem 3.. The form of the probability mass fuctio of geeralized Multiomial distributio is P(,,, ; a,, d, p, p,, p )!!! p p p!!! p p p,,, a Cosiderig such that + ad p + p, we obtai P(, ; a,, d, p, p ) > P( ; a,, d, p, p ) ()! p!! p ()!!! p p ()!! ( )! p ()!! ( )! p, a a p p a+d Lettig, p p ad p p p q, we have ()!! ( )! p q P(; a,, d, p) ()! a! ( )! p q p q > P(; a,, d, p) a p q Hece, P(; a,, d, p) p q p q a ; a, a + d, a + d,, (3.) which is the probability mass fuctio of geeralized Biomial distributio. 83