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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 iii their minimums. Geometric distribution, Exponential distribution, Stochastic ordering, Order statistics

The expectations E(X(sub 1)), E(Z(sub 1)), and E(Y(sub 1)) of the minimum of n independent geometric, modifies geometric, or exponential random variables with matching expectations differ. We show how this is accounted for by stochastic variability and how E(X(sub 1))/E(Y(sub 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 the minimums.