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On the minimum of independent geometrically distributed random variables. (English) Zbl 0830.62014
Summary: 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.

62E10 Characterization and structure theory of statistical distributions
62G30 Order statistics; empirical distribution functions
Full Text: DOI
[1] Margolin, B.H.; Winokur, H.S., Exact moments of the order statistics of the geometric distribution and their relation to inverse sampling and reliability of redundant systems, J. amer. statist. assoc., 62, 915-925, (1967)
[2] Ross, S.M., ()
[3] Trivedi, K.S., ()
[4] Uppuluri, V.R.R., A characterization of the geometric distribution, Ann. math. statist., 35, 1841, (1964), (Abstract 61)
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