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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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##### References:
 [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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