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The Largest Missing Value in a Sample of Geometric Random Variables

Published online by Cambridge University Press:  22 May 2014

MARGARET ARCHIBALD  and
ARNOLD KNOPFMACHER
MARGARET ARCHIBALD
Affiliation:
The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, PO Wits, 2050 Johannesburg, South Africa (e-mail: zigarch@gmail.com)
ARNOLD KNOPFMACHER
Affiliation:
The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, PO Wits, 2050 Johannesburg, South Africa (e-mail: arnold.knopfmacher@wits.ac.za)
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Abstract

We consider samples of n geometric random variables with parameter 0 < p < 1, and study the largest missing value, that is, the highest value of such a random variable, less than the maximum, that does not appear in the sample. Asymptotic expressions for the mean and variance for this quantity are presented. We also consider samples with the property that the largest missing value and the largest value which does appear differ by exactly one, and call this the LMV property. We find the probability that a sample of n variables has the LMV property, as well as the mean for the average largest value in samples with this property. The simpler special case of p = 1/2 has previously been studied, and verifying that the results of the present paper coincide with those previously found for p = 1/2 leads to some interesting identities.

Keywords

Primary 60C05Secondary 05A16
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 5: Honouring the Memory of Philippe Flajolet - Part 1 , September 2014 , pp. 670 - 685
Copyright
Copyright © Cambridge University Press 2014 

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