In this paper, the classes of Symmetrie density functions which depend on a skewness Parameter have been studied. In particular the skew normal, uniform, t, Cauchy, Laplace, and logistic distributions are given and some of their properties are explored.
In this work we investigate the generalized skew-symmetric distributions. Suppose Y is an absolutely continuous random variable symmetric about 0 with probability density function f and cumulative distribution function F. If a random variable X satisfies X 2 ¼ d Y 2 , then X is said to have a generalized skew distribution of F (or f). The generalized skew-Cauchy (GSC) distribution are considered and special examples of GSC distribution are presented. Some of these examples are generated from generalized skew-normal or generalized skew-t distributions. r 2007 Elsevier B.V. All rights reserved.MSC: primary 60E05; secondary 62E10
Let γ t
and δ t
denote the residual life at t and current life at t, respectively, of a renewal process , with the sequence of interarrival times. We prove that, given a function G, under mild conditions, as long as holds for a single positive integer n, then is a Poisson process. On the other hand, for a delayed renewal process with the residual life at t, we find that for some fixed positive integer n, if is independent of t, then is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t
and δ t.
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