Abstract
We consider samples of n geometric random variables W1 W2 ... Wn where P{W) = i} = pqi-l, for 1 ? j ? n, with p + q = 1. For each fixed integer d > 0, we study the probability that the distance between the consecutive maxima in these samples is at least d. We derive a probability generating function for such samples and from it we obtain an exact formula for the probability as a double sum. Using Rice's method we obtain asymptotic estimates for these probabilities. As a consequence of these results, we determine the average minimum separation of the maxima, in a sample of n geometric random variables with at least two maxima.
Highlights
We consider samples of n geometric random variables (ω1 ω2 · · · ωn) where P{ωj = i} = pqi−1, for 1 ≤ j ≤ n, with p + q = 1
Skip lists are an alternative to tries and digital search trees
A geometric random variable defines the number of pointers that it contributes to the data structure
Summary
We consider samples of n geometric random variables ω1 ω2 · · · ωn where P{ωj = i} = pqi−1, for 1 ≤ j ≤ n, with p + q = 1. For each fixed integer d > 0, we study the probability that the distance between the consecutive maxima in these samples is at least d. We derive a probability generating function for such samples and from it we obtain an exact formula for the probability as a double sum. Using Rice’s method we obtain asymptotic estimates for these probabilities. As a consequence of these results, we determine the average minimum separation of the maxima, in a sample of n geometric random variables with at least two maxima
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