Abstract
Cumulative Entropy (CE) as a measure of uncertainty alternative to Shannon entropy is proposed by Di Crescenzo and Longobardi (2009). In this paper, some properties of the cumulative entropy are derived and under conditions are showed the cumulative entropy of the last order statistics can determine the distribution function uniquely. Weibull family is characterized by ratio of the cumulative entropy of the last order statistics to its expectation. Also, some inequalities are presented for the cumulative entropy of reversed residual lifetime of a parallel system.
Highlights
Shannon (1948) introduced entropy, known Shannon entropy, as a measure of uncertainty for a random variable
Some properties of the cumulative entropy are derived and under conditions are showed the cumulative entropy of the last order statistics can determine the distribution function uniquely
Weibull family is characterized by ratio of the cumulative entropy of the last order statistics to its expectation
Summary
Shannon (1948) introduced entropy, known Shannon entropy, as a measure of uncertainty for a random variable. Shannon entropy h(X) of a continuous non-negative random variable X with probability density function f(x) is defined as. Rao et al (2004) defined Cumulative Residual Entropy(CRE) as an alternative measure of uncertainty to Shannon entropy in that the probability density function is replaced by survival(reliability) function and obtained some properties and applications of that(2005). The reversed residual lifetime is a concept in reliability that is convenient to describe the time elapsing between the failure of a system and the time when it is down This measure is defined as follows: CE(X) = − ∫ F(x) log F(x) dx (1). They proposed dynamic form of cumulative entropy that called dynamic cumulative entropy and obtained some of its properties
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