 Applied Mathematics, 2011, 2, 1382-1386 doi:10.4236/am.2011.211195 Published Online November 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Distribution of Geometrically Weighted Sum of Bernoulli Random Variables Deepesh Bhati1, Phazamile Kgosi2, Ranganath Narayanacharya Rattihalli1 1Department of Stat i s t i c s, Central University of Rajasthan, Kishangarh, India 2Department of Stat i s t i c s, University of Botswana, Gaborone, Botswana E-mail: dipesh089@gmail.com, KGOSIPM@mopipi.ub.bw, rnr5@rediffmail.com Received July 6, 2011; revised October 5, 2011; accepted October 13, 2011 Abstract A new class of distributions over (0,1) is obtained by considering geometrically weighted sum of independ-ent identically distributed (i.i.d.) Bernoulli random variables. An expression for the distribution function (d.f.) is derived and some properties are established. This class of distributions includes U(0,1) distribution. Keywords: Binary Representation, Probability Mass Function, Distribution Function, Characteristic Function 1. Introduction Uniform distribution plays an important role in Statistics. The existence of uniform random variable (r.v.) over the interval (0,1), using B(1,1/2) r.v.s is indicated in . As a generalization, in this paper we consider the following geometrically weighted sum of i.i.d. Bernoulli r.v.s 112jjjXZ, (1) where jZs are i.i.d. B(1,p) r.v’s. The remainder of the paper is organized as follows. In Section 2 we obtain the characteristic function of X and give an interpretation for the variable X. In Section 3 we derive the distribution function of X and prove some of its properties. In Section 4 we discuss the existence of the density function. In Section 5 distribution of sum of a finite number of vari-ables is considered and the graphs of its probability mass function (p.m.f.) and distribution function (d.f.) are given in the Appendix. 2. The Characteristic Function and an Application of the Model 2.1. The Characteristic Function Let ()FtPXt be the d.f. of X. Then by the defi-nition of X we have 11 11()001 11(1 ).222FtP XtZPZPXtZPZXXPtpP tp    (2) Hence the characteristic function (c.f.) ()t of X satisfies the equation 2()(1)( 2)e(2)ittptpt  . That is, 2()(2) 1eittt pp . Repeating this and replacing t by 2t each time, we get, for n = 1, 2,. 21()(2 )1e.knnitktt pp (3) The reproductive property of the characteristic func- tion exhibited by (3) is comparable to the characteristic function of an infinitely divisible distribution. For details one may refer to Section 7 of . Since (2)1 as ntn we have 21()1e kitktpp . (4) Note that if p = 0, this infinite product is 1, and, if p = 1, the infinite product is . Thus p = 0 results in X be- ing degenerate at 0 while p = 1 implies that X is degene- rate at 1. If p = 1/2, then the product term in (4) is eit21e 1e 12e1nit itnit it as n Thus if p = 1/2 then X has U(0,1) distribution. 2.2. An Application This resulting distribution can be used as a model in a D. BHATI ET AL.1383 1,situation similar to the following. Suppose a particle has linear movement on the interval [0,1]. To capture the particle suppose the following binary capturing tech-nique of dividing the existing interval into two equal halves is used. Suppose initially there are two barriers put at 0 and 1. After one unit of time a barrier is put at the midpoint of 0 and 1. Further the interval in which the particle is found is divided into two equal halves by placing a barrier at their midpoint and the process is con-tinued. The intervals containing the particle keep on shrinking and finally shrink to X, the point at which the particle is captured in the long run. The behavior of the particle is known only to the extent that at the moment of placing a barrier after exactly one unit of time the parti-cle is on the right side of the inserted barrier with prob-ability p. 3. Main Results 3.1. Notation It is known that every number t, has a binary representation through as 0t,0or11nnaa112iiita. If a number t has the representation 11, 1,2ikikita a we refer t as a finite binary termi- nating number and such a number can be represented by for some . However such a number also can be represented as 2,kr1, 2,,21kr11,,1,2,,20,1,1, 2,iiiiikibbai kbbikk  1, It is to be noted that the right tail of the sequence {ai} is of the form (0,0,0 while that of the sequence {bi} is of the form (1,1,1, In the following as a matter of convention we do not consider representation with the right tail of the form (1,1,1,). Under such a convention, corresponds to a unique binary sequence )).[0,1]t1ia and conversely. If 112iiita, then we shall denote this relation as , (BR to mean the binary representation). BRnta 3.2. Properties Theorem 1: Let and BRnta1kkjjsa, . Then the d.f. of X defined in (1) is given by 01, 2,,0ks111() jjjs sjjFtP Xtaqp. (5) Proof: Let, 112iiita and r be the rth non-zero element kain the sequence ,1,2,iar. Let be the number having the binary representation 12 rk and 0rt,aa(,, 0,0,)a0t. It is to be noted that r is a finite binary terminating number. If t is not a finite binary terminating number then the sequence increases to t. If t is a finite binary terminating number then we note that rtttrt for some finite r. For example, let have a binary rep-resentation 00010011000 and t12344,7, 8, 11kkkk so that 1BRt (001000, 20)BRt (0001100300 0), BRt (000100110 so on. We note that 1ia00 ) and for ,,ik k and 0 12 3,kotherwise. Note that PXt,1, 2,,1,1,0,1, 2,...1,0,1, 2,0.    rrrrriir kkjkkjPZ aik ZZjPZ Zj Thus F does not have a jump at tr and rFtFt. ng number. WLet t be not a finite binary representie note that 111100,,kkXtPZ Zq P011122121 1110, ,0,1,0,,0kkkkkPtX tPZZZZZqp  1113222 323 11122110, ,0,1,0,,0,1,0,,0kkkkkkk kPtX tPZZZZ.ZZZZ qp   Thus in general for r = 1, 2, 3, we have (1)1rr 1.rkrrPtX tqp  Hence .(1)1111() rkr rrrrrPX tPtXtqp  ,rkjIf we let then s, then we will have In fact F does not have jump at ant t, (0 < t < 1). Hence 1111jijira 11.jjjs sq p 1jja()PX t111.jjjs sjjPX tPXtaqp  Copyright © 2011 SciRes. AM D. BHATI ET AL. 1384 t is a finite binary termination number th for some finite number r and since articular Case However, if en ttr rrjs stPXtPX t 11 1111jj jjjs srj jjjPXPXta qpaqp   Theorem 4: For 0,1, 2,jajrr. 3.3. P If 1p, then 211,(0,1)2jjjPXt att . ct it can befied that (5) satisfies (2). It fol- fact that if t has binary representation a = (aHence ~(0Remks: ,1)XU . arIn Fa verilows from the1, a2,) then for 1) 0 < t < 1, t/2 has binary representation 12(, ,)uuu where u= 0 and ui+1 = a,i = 1, 2, and 1i tation 1 2an, 0, 2n rpectithen tTheorem 3: with 2) 1/2 < t < 1, 21t has binary represenrv =12,,vv whee vi = ai+1, i = 1, 2,. Theorem 2: If u and vhave binary representations (a, a,,0, , a,,a, 1, 1,,1) es,0) and (a1vely he2 conditional distribution of X given u ≤ X ≤ v is that of nuX. Proof: Follows by the definition of X. RLet Bta with 12,,,,0,0,kaa an1ka. Then for 10s2k 10P 0022kkXtsXt PXs  . Proof: 102ts we have 1122qPtXs PtXsp . Proof: Let (2)1123,,,ZZZZZZ  ,(2)1(2) (2)10,0Pt XsPZZAPZPZ AqPZ A   and (2)111 1,22PtX sPZZA  (2) (2)11.PZ ApPZ A In the above in fact PZ (2) .2XPZAPst  Hence the result. 5: Theorem(.,If )Fp and are the diribution functions of respen If (., )Gp ectively thst X an)d 1-XProof: (,(,1)Gxp Fxp. 112jjjXZ and jZ’s are i.i.d. ),1( pB then 11111122jj1122111221122011220020002021,,,,2,,,11,,, 22,,,kkkjkk jjkkkkjkk jjPXtsXtPXtsPXtstt PXtPZ aZaZaZsPZ aZaZaPZa ZaZaPZsPZ aZa       12.2kkkkZaPXsPXs jjjjZY    X where Yj’s are i.i.d. (1,1 )Bp. Hence the result. Mean and Va ri an ce of X: 1j11() 22jjjjEXEZpp 1  2211212111212() 2111 22211 213221(1 2      jij11112( )2  i1233jijjE EZZijjijijiiiiX EZpppppppj 2)3pHence (1 )() .ppVar X 3By using the c.f. 21()1e kitktpp  the cu-mulants can also be obtained. Copyright © 2011 SciRes. AM D. BHATI ET AL. Copyright © 2011 SciRes. AM 13854. Nonexistence of Density Function We have proved that the distribution function of X is given by Let the left derivative and the right derivative of F at t exist. These be denoted by111() jjjs sjjFta qp, BRnta. ()ft and ()ft respectivelyConsider . 12f and 12f, the right and left de-rivatives of F at 1/2. 111111222lim2 lim2kkk 2lim(2 )(2)kkkkFFfqp qp   kand 11limlim(2)(2 ).2kkkkpq pq 11 122kkkat 1222limkFFf  Note th12f and 12f are equal if and only if 1pq. He2t differentiable at 1/2 if nce F is no12p. Let :1,2,3,0,1,2,,22kkrDk r. It can be that F is not differentiable at each point of D and D is countable dense subset of [0,1]. Hence F is nowhere differentiable in [0,1]. 5. Distribution of Sum of a Finite Numbr of Bernoulli Random Variables Since F(t) is an infinite series for t not in D, the exact eva- lu is not pe for eacing we se-over, the density function of X does not exist on the in-terval (0,1). Hence in the follow consider thequence {Xk} of r.v. defined by shown the set eation of F(t)ossiblh (0,1).t More-11.2ikkiiXZ (6) The sequence kX increases point wise to X and 2.kkXX Similar to (5), it can be shown that for 112jjjta, k111()( )jjjs skjjFtaqpFthe d.f. of Xk is s, whjere 11.2kjsaFurther, at these values of s, j() ()k FsFs and vathe difference between two successivelues of s’s is 2k, as such the two functions ()kFt and ()Ft are almost alike. Siiso X the nce the sequensequence {Fk(t)} dece {Xk} creasesincreases point we t to F(t) for (0, 1)t. kWe note that for p = 1/2, ()( )FtFs. The gra5phs of p.m.f anX10 for different values of d d.f. of X and 12p are gi H. J. Vama of the paper. . References ] S. Kunte and R. N. Rattihalli, “Uniform Random Vari-n, Academic Press, Cambridge, 2001. ven in the Appendix. 6. Acknowledgements We are thankful toor Professn, Central Uni-versity of Rajasthan, India, for the discussions which helped to improve the content and the presentation 7 [1able. Do They Exist in Subjective Sense?” Calcutta Sta-tistical Association Bulletin, Vol. 42, 1992, pp. 124-128.  K. L. Chung, “A Course in Probability Theory,” 3rd Edi-tio D. BHATI ET AL. 1386 Appendix The graphs of p.m.f and d.f. of Xk for different values of p. Probability mass function Distribution function k = 5 k = 10 k = 5 k = 10 p = 0.3 p = 0.3 p = 0.7 p = 0.7 p = 0.9 p = 0.9 Copyright © 2011 SciRes. AM