On the minimum of independent geometrically distributed random variables

Gianfranco Ciardo, Lawrence M. Leemis, David Nicol

Research output: Contribution to journalArticlepeer-review


The expectations E[X(1)], E[Z(1)], and E[Y(1)] of the minimum of n independent geometric, modified geometric, or exponential random variables with matching expectations differ. We show how this is accounted for by stochastic variability and how E[X(1)]/E[Y(1)] equals the expected number of ties at the minimum for the geometric random variables. We then introduce the "shifted geometric distribution", and show that there is a unique value of the shift for which the individual shifted geometric and exponential random variables match expectations both individually and in their minimums.

Original languageEnglish (US)
Pages (from-to)313-326
Number of pages14
JournalStatistics and Probability Letters
Issue number4
StatePublished - Jun 1995
Externally publishedYes


  • Exponential distribution
  • Geometric distribution
  • Order statistics
  • Stochastic ordering

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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