Abstract
The discrete Shannon entropy H was formulated only to measure indeterminacy effected through a set of probabilities, but the indeterminacy in a real-valued discrete variable depends on both the allowed outcomes x and the corresponding probabilities Þ. A fundamental measure that is sensitive to both x and p is derived here from the total differential entropy of a continuous real variable and its conjugate in the discrete limit, where the conjugate is universally eliminated. The asymptotic differential entropy recovers H plus the new measure, named ≡, which provides a novel probe of intrinsic organization in sequences of real numbers.
Highlights
Let Y be a discrete variable with n > 1 generic outcomes y = {y1, . . . , yn } and corresponding probabilities p = {p1, . . . , pn } such that n pj = 1 (1)
Expressed in terms of the natural logarithm, the Shannon entropy attributed to Y in connection with p is [1]
Shannon entropy was formulated only to measure indeterminacy effected by a given p in a generic discrete variable
Summary
The set of probabilities governing Xre is of the form pr , which is defined generally to be indistinguishable from n randomly selected positive real numbers, subsequently normalized to unity and listed in random order. The outcomes of Xae are matched to a set of probabilities of the form pa , which is defined generally to be indistinguishable from n randomly selected positive real numbers, subsequently normalized to unity and listed in ascending order. Shannon entropy was formulated only to measure indeterminacy effected by a given p in a generic discrete variable. The purpose of this article is to demonstrate that a measure of indeterminacy for discrete real variables that is sensitive to both x and p emerges from the total differential entropy of XX and its conjugate in the limit as ν diverges.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
