Electronic copy available at: http://ssrn.com/abstract=2343921 A note on double truncated (interval) weighted cumulative entropies S. Yasaei Sekeh ∗ , G. R. Mohtashami Borzadran † , A. H. Rezaei Roknabadi ‡ Abstract. Measure of the weighted cumulative entropy about the predictability of failure time of a system have been introduced in [3]. Referring properties of doubly truncated (interval) cumulative residual and past entropy, several bounds and properties in terms of the weighted cumulative entropy is proposed. 2000 MSC. 62N05, 62B10 Keywords: weighted cumulative entropy, double truncated (interval) weighted cumulative (residual) entropy, weight function. 1 Introduction. Interval weighted cumulative entropies Let x ∈ R + → φ(x) ≥ 0 be a given measurable function. The weighted cumulative residual entropy (WCRE) E w φ (X ) and the weighted cumulative entropy (WCE) E w φ (X ) of a RV X with a cumulative distribution function (CDF) F and survival function (SF) ¯ F are defined by E w φ (X )= E w φ (F )= −  R + φ(x) ¯ F (x) log ¯ F (x)dx, and (1.1) E w φ (X )= E w φ (F )= −  R + φ(x)F (x) log F (x)dx, (1.2) respectively. Assume that all integrals are absolutely convergent with the standard agreement 0 log 0 = 0 log ∞ = 0. Cf. [8], [3], [1] and [6]. * Department of Statistics, Federal University of S˜ ao Carlos (UFSCar), S˜ ao Carlos, Brazil. E-mail: sa-yasaei@yahoo.com † Department of Statistics, Ferdowsi University of Mashhad, Mashhad, Iran. E-mail: gmb1334@yahoo.com; grmohtashami@um.ac.ir ‡ Department of Statistics, Ferdowsi University of Mashhad, Mashhad, Iran. E-mail: rezaei494@gmail.com 1