TY - JOUR

T1 - On the minimum of independent geometrically distributed random variables

AU - Ciardo, Gianfranco

AU - Leemis, Lawrence M.

AU - Nicol, David

N1 - Funding Information:
This research was partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS 1-19480 while the authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681. * Corresponding author. E-mail: {ciardo, leemis, nicol} @cs.wm.edu.

PY - 1995/6

Y1 - 1995/6

N2 - 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.

AB - 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.

KW - Exponential distribution

KW - Geometric distribution

KW - Order statistics

KW - Stochastic ordering

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U2 - 10.1016/0167-7152(94)00130-Z

DO - 10.1016/0167-7152(94)00130-Z

M3 - Article

AN - SCOPUS:33645570891

SN - 0167-7152

VL - 23

SP - 313

EP - 326

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

IS - 4

ER -