On the minimum of independent geometrically distributed random variables

Abstract

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.