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 SM ISO690:2012ZBĂGANU, Gheorghiță, RADUCAN, Anisoara Maria. What is the chance to have a leader in a random set? In: Conference on Applied and Industrial Mathematics: CAIM 2022, Ed. 29, 25-27 august 2022, Chişinău. Chișinău, Republica Moldova: Casa Editorial-Poligrafică „Bons Offices”, 2022, Ediţia a 29, p. 122. ISBN 978-9975-81-074-6. |
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| Conference on Applied and Industrial Mathematics Ediţia a 29, 2022 |
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Conferința "Conference on Applied and Industrial Mathematics" 29, Chişinău, Moldova, 25-27 august 2022 | ||||||
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Let us consider a population of individuals characterized by the same set of features. We denote the population by (Zi)i≥1 and we assume that these are d-dimensional (d ≥ 2) independent, identically distributed random vectors. Let us select a finite sample: Sn = {Z1,Z2, ..,Zn}, n ≥ 1. If there exists Zj ∈ Sn such that Zj ≥ Zk, ∀1 ≤ k ≤ n, then we say that Zj is a leader of Sn. If there exists Zj ∈ Sn such that Zj ≤ Zk, ∀1 ≤ k ≤ n, then we dename Zj an anti-leader of Sn.Here the comparison of two vectors has the usual sense: if Zj = Z(j) 1 ,Z(j) 2 , ...,Z(j) d and Zk = Z(k) 1 ,Z(k) 2 , ...,Z(k) d then Zj ≥ Zk ⇔ Z(j) i ≥ Z(k) i , ∀1 ≤ i ≤ d. Our purpose is to compute (if possible) or to estimate the probability that a leader ( or an anti-leader, or both) does exist in a given sample. We focus our study on a particular case: precisely we consider Z =f (X) with f = (f1, f2, ..., fd) : [0, 1] → Rd, X a random variable uniformly distributed on [0, 1] and, in most examples, f1 (X) = X. |
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