On the Distribution of the Minimum and of the Maximum of a Random Number of I.I.D. Random Variables
Let Xi, i = 1,2,… be i.i.d. random variables with a common distribution function F(x). Let N be a positive integer-valued random variable, independent of the X’s, with a probability generating function ψ(u). Then G(x) = ψ(F(x)) and H(x) = 1−ψ(1−F(x)) are, respectively, the distributions of max(X1,…,XN) and of min(X1,…,XN). In sections 2 and 3 we derive some inequalities concerning F, G and H, their densities and their hazard rates. We also discuss there some cases in which the monotonicity of the hazard rate of F is preserved by G and H. In section 4 we give some solutions of a functional equation that appears as a condition in some of the theorems of section 3. We also characterize the geometric distributions by requiring the solutions to satisfy an additional assumption. By taking Xi and/or N as random vectors, we introduce in section 5 some methods of construction of multivariate distributions with desired marginals.
Key WordsSeries and parallel systems hazard rate random walks geometric distribution Jensen’s inequality functional equation multivariate distributions with desired marginals log-convex functions
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