Computational & Applied Mathematics & Statistics
Statistics and Probability Letters
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.
Ciardo, Gianfranco; Leemis, Lawrence; and Nicol, David, On the minimum of independent geometrically distributed random variables (1995). Statistics and Probability Letters, 23(4), 313-326.
This version is the accepted (post-print) version of the manuscript.