Starting with the second Lagrange expansion, with f(z) and g(z) as two probability generating functions defined on nonnegative integers such that g(0) 0, we define and study a new class of discrete probability distributions called the Lagrange distributions of the second kind. This class has the probability function:z=O for y =0, 1, 2,.... Different families are generated by various choices of the functions f(z) and g(z). Families of the weighted distributions that correspond to the Lagrange distributions of the first kind are defined using different weight functions. For a particular form of the weight function, it is shown that, under some conditions, the weighted distributions of the members of the Lagrange distributions of the first kind belong to the class of the Lagrange distributions of the second kind. Weighted distributions of the Borel-Tanner distribution, Haight's distribution, generalized Poisson and generalized negative binomial distributions, etc., are shown to be the members of the class of Lagrange distributions of the second kind. Certain properties of the Lagrange distributions of the second kind are given.
It is well-known that the inequalities used in the definition of the New Better than Used (N.B.U.) and the New Better than Used in Expectation (N.B.U.E.) concepts, see BARLOW and PROSCHAN (1965, 1975) become equalities if, and only if, the life length of an organism follows an exponential distribution. I t is proved in the present paper that these inequalities also reduce to equalities for the class of life distributions that have the "setting the clock back to zero" property. Simple examples of these distributions include the exponential, the linear hazard exponential and the Gompertz distributions. The General Krane distributions (Krane 1963) belong to this class, as well as a recent model introduced by CHIANG and CONFORTI (1989) of a survival distribution in which the hazard rate is a function of the accumulated effect of an individual's continuous exposure to the toxic material in the environment and his biological reaction to the toxin absorbed.As a simple application of the result proved in the paper, the life expectancy of an organism at age 1, involved in the N.B.U.E. concept is evaluated for the Gompertzian growth process and for the Chiang and Conforti model.
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