Distribution of Geometrically Weighted Sum of Bernoulli Random Variables

where j Z s  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 variables is considered and the graphs of its probability mass function (p.m.f.) and distribution function (d.f.) are given in the Appendix.


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 [1].As a generalization, in this paper we consider the following geometrically weighted sum of i.i.d.Bernoulli r.v.s 1 1 2 where j Z s  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 variables is considered and the graphs of its probability mass function (p.m.f.) and distribution function (d.f.) are given in the Appendix.

The Characteristic Function and an
Application of the Model

The Characteristic Function
Let  ( )


F t P X t   be the d.f. of X.Then by the definition of X we have Repeating this and replacing t by 2 t each time, we get, for n = 1, 2, .
The reproductive property of the characteristic function exhibited by (3) is comparable to the characteristic function of an infinitely divisible distribution.For details one may refer to Section 7 of [2].

Since
( 2 ) 1 as Note that if p = 0, this infinite product is 1, and, if p = 1, the infinite product is .Thus p = 0 results in X being degenerate at 0 while p = 1 implies that X is degenerate at 1.If p = 1/2, then the product term in ( 4) is as n   Thus if p = 1/2 then X has U(0,1) distribution.

An Application
This resulting distribution can be used as a model in a 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 technique 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 continued.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 particle is on the right side of the inserted barrier with probability p.

Notation
It is known that every number t, has a binary representation through as we refer t as a finite binary terminating number and such a number can be represented by for some .However such a number also can be represented as It is to be noted that the right tail of the sequence {a i } is of the form (0,0,0 while that of the sequence {b i } 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 ) ). [ , then we shall denote this relation as , (BR to mean the binary representation).

 
BR n t a 

Properties Theorem 1:
Let and Thus F does not have a jump at t r and     r F t F t  .ng number.W Let t be not a finite binary representi e note that , 0 Thus in general for r = 1, 2, 3 , we have If we let then s , then we will have In fact F does not have jump at ant t, (0 < t < 1). Hence t is a finite binary termination number th for some finite number r and since articular Case ( , , ) u u u   where u = 0 and u i+1 = a ,i = 1, 2, and

P
Then for In the above in fact . 2 where Y j 's are i.i.d.
Hence the result.

Mean and Variance of X:
  By using the c.f.

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 , the right and left de-

Distribution of Sum of a Finite Numb r of Bernoulli Random Variables
Since F(t) is an infinite series for t not in D, the exact evalu is not p e for eac ing we se-over, the density function of X does not exist on the interval (0,1).Hence in the follow consider the quence {X k } of r.v.defined by shown the set Similar to (5), it can be shown that for .References ] S. Kunte and R. N. Rattihalli, "Uniform Random Varin, Academic Press, Cambridge, 2001.ven in the Appendix.

Acknowledgements
We are thankful to or Profess n, Central University of Rajasthan, India, for the discussions which helped to improve the content and the presentation