Metrika (2006) 64: 47–61 DOI 10.1007/s00184-006-0030-6 ORIGINAL ARTICLE Asok K. Nanda · Prasanta Paul Some properties of past entropy and their applications Received: 5 May 2004 / Revised: 15 February 2005 / Published online: 21 February 2006 © Springer-Verlag 2006 Abstract In the context of information theory, measure of uncertainty in past lifetime distribution has been proposed by Di Crescenzo and Longobardi (J Appl Probab 39:434–440, 2002). In this paper, we study some ordering and aging prop- erties in terms of past entropy (based on past lifetime) and develop some charac- terization results. Some discrete distribution results are also addressed here. Keywords DRHR · LU and PE orders · Past entropy · Residual entropy · Reversed hazard rate function · Reversed mean residual function AMS Subject Classification (2000) 94AXX · 94A17 1 Introduction Let X be an absolutely continuous nonnegative random variable having distribu- tion function F (t ) = P ( X ≤ t ) and survival function ¯ F (t ) = P ( X > t ). X may be the lifetime of a component/system or of a living organism. The basic measure of uncertainty is defined by Shannon (1948) as H ( f ) =- ∞  0 f (x ) ln f (x )dx =-E (ln f ( X )), (1.1) where f (x ) is the density function of X . This is known as Shannon information measure. The properties and virtues of H ( f ) have been thoroughly investigated A.K. Nanda (B ) · P. Paul Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, India E-mail: asok@maths.iitkgp.ernet.in E-mail: ppaul@maths.iitkgp.ernet.in