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A Bivariate Uniform Distribution

In: Contributions to Probability and Statistics

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  • Albert W. Marshall

    (University of British Columbia)

Abstract

The univariate distribution uniform on the unit interval [0,1] is important primarily because of the following characterization: Let X be a random variable taking values in [0,1]. Then the distribution of X + U (mod 1) is the same as the distribution of X for all nonnegative random variables U independent of X if and only if X has a distribution uniform on [0,1]. A natural bivariate version of this is the following: Let (X,Y) be a random vector taking values in the unit square. Then (*) (X + U (mod 1), Y + V (mod 1)) has the same distribution as (X,Y) for every pair (U,V) of nonnegative random variables independent of (X,Y) if and only if X and Y are independent and uniformly distributed on [0,1]. But if (*) is required to hold only when U = V with probability one, then (X,Y) can have any one of a large class of bivariate uniform distributions which are given an explicit representation and studied in this paper.

Suggested Citation

  • Albert W. Marshall, 1989. "A Bivariate Uniform Distribution," Springer Books, in: Leon Jay Gleser & Michael D. Perlman & S. James Press & Allan R. Sampson (ed.), Contributions to Probability and Statistics, chapter 6, pages 99-106, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-3678-8_9
    DOI: 10.1007/978-1-4612-3678-8_9
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