On the Minimum of Independent Geometrically Distributed Random Variables

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