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On the Modular Value and Fractional Part of a Random Variable

Published online by Cambridge University Press:  27 July 2009

Stephen J. Herschkorn
Stephen J. Herschkorn
Affiliation:
School of Business and RUTCOR, Rutgers University, New Brunswick, New Jersey 08903
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Abstract

Let X be a random variable with characteristic function ϕ. In the case where X is integer-valued and n is a positive integer, a formula (in terms of ϕ) for the probability that n divides X is presented. The derivation of this formula is quite simple and uses only the basic properties of expectation and complex numbers. The formula easily generalizes to one for the distribution of X mod n. Computational simplifications and the relation to the inversion formula are also discussed; the latter topic includes a new inversion formula when the range of X is finite.

When X may take on a more general distribution, limiting considerations of the previous formulas suggest others for the distribution, density, and moments of the fractional part X — [X]. These are easily derived using basic properties of Fourier series. These formulas also yield an alternative inversion formula for ϕ when the range of X is bounded.

Applications to renewal theory and random walks are suggested. A by-product of the approach is a probabilistic method for the evaluation of infinite series.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 9 , Issue 4 , October 1995 , pp. 551 - 562
Copyright
Copyright © Cambridge University Press 1995

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