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.