A characterization of geometric distributions by distributional properties of order statistics

Abstract
Let X 1, X 2 be independent identically distributed positive integer valued random variables. H the X i 's have a geometric distribution, then the conditional distribution of R = max(X 1, X 2)-min(X 1, X 2), given R > 0, is the same as the distribution of X 1. This property is shown to characterize the geometric distribution.

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