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1. INTRODUCTION
Let X be a continuous random variable with probability density function (p.d.f.) f (x ), distribution function F (x ), and survival function F (x ) = 1 - F (x ). We define the p.d.f. f *(x )as
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where Μ = E (X ) < ∞. Then f *(x ) is called the p.d.f. of an equilibrium distribution or induced distribution.
The above distribution arises as the limiting distribution of the forward recurrence time in a renewal process. It also arises as the marginal distribution of W1 , where the joint p.d.f. of (W1 ,W2 ) is given by
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see Brown [10].
Note that, in this case, the p.d.f. of W2 is given by the length-biased version of the original distribution as
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The equilibrium distribution (1.1) is intimately connected to its parent distribution and many of the reliability properties of the original distribution can be easily studied by means of the properties of the equilibrium distribution.
The purpose of this article is to study the relationships between (1.1) (including its higher derivatives) and the original distribution. Some stochastic order relations and the relations between their aging properties are investigated and some applications in the field of insurance and financial investments are given. This is primarily a review article. However, the examples in Section 6 are new.
The organization of this article is as follows. In Section 2 we present some definitions and background material encountered in reliability studies, including some criteria of aging and their relationships. Some definitions of stochastic order relations and their relationships are also provided. Section 3 deals with higher-order equilibrium distributions and stop loss moments. In Section 4 aging properties of equilibrium distributions and their stochastic ordering with the original distribution are explored. In Section 5 the relation between the equilibrium distribution of a series system and a series system of equilibrium distributions, consisting of two components, is investigated. Section 6 contains the bivariate equilibrium distribution along with two examples. Finally, in Section 7 we provide some conclusions and comments.
2. DEFINITIONS AND BACKGROUND
Let X be a continuous...