Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T01:58:45.593Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Drake, A.W. (1967). Fundamentals of applied probability theory. New York: McGraw-Hill.Google Scholar
2.Feller, W. (1971). An introduction to probability theory and its applications, Vol. II. New York: John Wiley and Sons.Google Scholar
3.Katznelson, Y. (1976). An introduction to harmonic analysis, 2nd corrected ed. New York: Dover.Google Scholar
4.Körner, T.W. (1988). Fourier analysis. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
5.Mardia, K.V. (1972). Statistics of directional data. London: Academic Press.Google Scholar
6.Oppenheim, A.V. & Shafer, R.W. (1975). Digital signal processing. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
7.Papoulis, A. (1962). The Fourier integral and its applications. New York: McGraw-Hill.Google Scholar
8.Rees, C.S., Shah, S.M., & Stanojević, C.V. (1981). Theory and applications of Fourier analysis. New York: Marcel Dekker.Google Scholar
9.Schatte, P. (1973). Zür Verteilungen der Mantisse in der Gleitkommadarstellung einer Zufallsgröβe. Zeitschrift für Angewandte Mathematik und Mechanik 53: 553–565.CrossRefGoogle Scholar
10.Schatte, P. (1983). On sums modulo 2π of independent random variables. Mathematische Nachrichten 110: 243–262.CrossRefGoogle Scholar
11.Stadje, W. (1984). Wrapped distributions and measurement errors. Metrika 31: 303–317.Google Scholar
12.Zygmund, A. (1959). Trigonometric series, Vol. I, 2nd ed.Cambridge: Cambridge University Press.Google Scholar