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## Abstract:To any discrete probability distribution P we can associate its entropy H(P ) = − p i ln p i and its index of coincidence IC(P ) = p 2 i . The main result of the paper is the determination of the precise range of the map P (IC(P ), H(P )). The range looks much like that of the map P (P max , H(P )) where P max is the maximal point probability, cf. research from 1965 (Kovalevskij [18]) to 1994 (Feder and Merhav [7]). The earlier results, which actually focus on the probability of error 1 − P max rather than P … Show more

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##### Cited by 65 publications
(97 citation statements)
##### References 22 publications
(33 reference statements)
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“…The problem of finding the minimum and maximum of the Shannon entropy under assumption that the index of coincidence is constant has been analyzed in [57] (some generalizations and related topics can be found also in [16,137]). By [57, Theorem 2.5.…”
Section: Remarkmentioning
confidence: 99%
“…The problem of finding the minimum and maximum of the Shannon entropy under assumption that the index of coincidence is constant has been analyzed in [57] (some generalizations and related topics can be found also in [16,137]). By [57, Theorem 2.5.…”
Section: Remarkmentioning
confidence: 99%
“…Actually, this problem is equivalent to seek the maximal and minimal H( µ) = − i µ i log 2 µ i for a given τ = 2(1 − i µ 2 i ), and the later one is a classical problem which has been solved in Refs. [48,49]. We use…”
Section: A Calculation Of Fmentioning
confidence: 99%
“…In Harremoës and Topsøe [1] the exact range of the map P (IC(P ), H(P )) with P varying over either M 1 + (n) or M 1 + (N) was determined. The ranges in question, termed IC/H-diagrams, were denoted ∆, respectively ∆ n :…”
Section: H(pmentioning
confidence: 99%
“…Figure 1 illustrates the situation for the restricted diagrams ∆ n . The key result of [1] states that ∆ n is the bounded region determinated by a certain Jordan curve determined by n smooth arcs, the "upper arc" connecting Q 1 and Q n and then n − 1 "lower arcs" connecting Q n with Q n−1 , Q n−1 with Q n−2 etc. until Q 2 which is connected with Q 1 .…”
Section: H(pmentioning
confidence: 99%
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