# Some topics in probability and statistics

#### Abstract

This dissertation consists of a series of four papers, each appearing as a separate chapter. In the first chapter we consider the expected number of i.i.d. samples required to see all of the balls in an urn. Suppose an urn contains m white balls, numbered 1,...,m, and suppose K1,K2... is a sequence of independent random variables with a common distribution on the positive integers. At time i we take a random sample of Ki [special characters omitted] m balls, paint them red, and return them to the urn. Let X be the number of such samples required to paint all of the balls in the urn red. We generalize the inclusion/exclusion argument used by Pólya (1930) for the fixed sample size case and derive an approximation to EX that is, in most situations, very accurate. The approximation derived agrees with that of Sellke (1995), who used a Walsd's identity and a Markov chain coupling technique to examine this problem. We then derive bounds on the error of the approximation in the case where the Ki are bounded. We conclude the chapter by deriving closed forms for EX in two special cases. In chapter two we present efficient procedures for generating random exponential and normal deviates based on the acceptance-complement method (Kronmal and Peterson, 1981). We provide comparisons with the corresponding Ziggurat procedures proposed by Marsaglia and Tsang (1984, 2000). The proposed procedures maintain good precision over the entire support of the respective densities and are very easy to set up and implement. The proposed exponential procedure compares favorably with the Ziggurat procedure in terms of speed, running up to 24% faster on some platform/compiler/uniform generator combinations tested. In chapter three we discuss optimal and approximately optimal fixed cost Bayesian sampling designs for simultaneous estimation in independent homogeneous Poisson processes. We develop general allocation formulae for a basic Poisson-Gamma model and compare these with more traditional allocation methods. We then discuss techniques for finding representative gamma priors under more general hierarchical models and show that, in many practical situations, these provide reasonable approximations to the hierarchical prior and Bayes' risk and hence may provide reasonable approximations to the allocation problem. The methods developed are general enough to apply to a wide variety of models and are not limited to Poisson Processes. In chapter four, the distribution of increasing 2-sequences in random permutations of the first n integers is generalized to random permutations of arbitrary multi-sets using a finite Markov chain embedding technique. A numerical example is provided to aid in understanding and some applications are briefly discussed.

#### Degree

Ph.D.

#### Advisors

Sellke, Purdue University.

#### Subject Area

Statistics

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