We show that the Poisson–Dirichlet distribution is the distribution of points in a scale-invariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T , and conditioning on the event that T 6 1. Restricting both processes to (0, β] for 0 < β 6 1, we give an explicit formula for the ...

We explore negative dependence and stochastic orderings, showing that if an integer-valued random variable W satisfies a certain negative dependence assumption, then W is smaller (in the convex sense) than a Poisson variable of equal mean. Such W include those which may be written as a sum of totally negatively dependent indicators. This is generalised to other stochastic orderings. Application...

  This expository article shows how the maximum likelihood estimation method and the Newton-Raphson algorithm can be used to estimate the parameters of the power-law Poisson process model used to analyze data from repairable systems .