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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.

MSC:
62E10 Characterization and structure theory of statistical distributions
62G30 Order statistics; empirical distribution functions
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[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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