Finding the distribution of a statistic is always an important problem that we face in statistical inference. Methods that are usually used for solving this problem are change of variable technique, distribution function technique and moment generating function technique. Among these methods change of variable technique is the most commonly used one. This method is simple when the statistic is a one-to-one transformation of the sample observations and if it is many-to-one, then one needs to compute the jacobian for each partition of the range for which the transformation is one-to-one. In addition, if we want to find the distribution of a statistic involving n random variables using the change of variable technique, we have to define (n-1) auxiliary variables. Unless these (n-1) auxiliary variables are carefully chosen, calculation of jacobian as well as finding the range of integration to obtain the marginal distribution of the statistic of interest become complicated. [See [3]]Au, Chi and Tam, Judy [1] proposed an alternative method of finding the distribution of a statistic by using Dirac generalized function. In this study we considera number of problems involving different probability distributions that are not quiet easy to solve by change of variable technique. We will illustrate the method by solving problems which include finding the distributions of sums, products, differences and ratios of random variables. The main purpose of the thesis is to show that using Dirac generalized function one can solve these problems with more ease. This alternative approach would be more suitable for students with limited mathematical background. / Department of Mathematical Sciences

Consider a probability distribution on the permutations of n elements. If the probability of each permutation is proportional to θK, where K is the number of cycles in the permutation, then we say that the distribution generates a θ-biased random permutation. A random permutation is a special θ-biased random permutation with θ = 1. The mth moment of the rth longest cycle of a random permutation is Θ(nm), regardless of r and θ. The joint moments are derived, and it is shown that the longest cycles of a permutation can either be positively or negatively correlated, depending on θ. The mth moments of the rth shortest cycle of a random permutation is Θ(nm−θ/(ln n)r−1) when θ < m, Θ((ln n)r) when θ = m, and Θ(1) when θ > m. The exponent of cycle lengths at the 100qth percentile goes to q with zero variance. The exponent of the expected cycle lengths at the 100qth percentile is at least q due to the Jensen’s inequality, and the exact value is derived.

The heat content of a domain D of &Ropf;^d is defined as</sp> < p >E(s) = &int;_D u(s,x)dx, where u is the solution to the heat equation with zero initial condition and unit Dirichlet boundary condition. This thesis studies the behaviour of E(s) for small s with a particular emphasis on the case where $D$ is a planar domain whose boundary is a random Koch curve. When &part;D is spatially homogeneous, we show that we can recover the upper and lower Minkowski dimensions of &part;D from E(s). Furthermore, in some cases where the Minkowski dimension does exist, finer fluctuations can be recovered and the heat content is controlled by s^&alpha; exp{f (log(1/s)} for small s, for some positive &alpha; and some regularly varying function f. When &part;D is statistically self-similar, the heat content asymptotics are studied using a law of large numbers for the general branching process, and we show that the Minkowski dimension and content of &part;D exist and can be recovered from E(s). More precisely the heat content has an almost sure expansion E(s) = c₁ s^&alpha; N_&infin; + o(s^&alpha;), a.s. for small s, for some positive c₁ and &alpha; and a positive random variable N_&infin; with unit expectation. To study the fluctuations around these asymptotics, we prove a central limit theorem for the general branching process. The proof follows a standard Taylor expansion argument and relies on the independence built into the general branching process. The limiting distribution established here is reminiscent of those arising in central limit theorems for martingales. When &part;D is a statistically self-similar Cantor subset of &Ropf;, we discuss examples where we have and fail to have a central limit theorem for the heat content. We conclude with an open question about the fluctuations of the heat content when &part;D is a statistically self-similar Koch curve.

This dissertation develops, applies, and investigates new methods to improve the analysis of logistic regression mixture models. An interesting dose-response experiment was previously carried out on a mixed population, in which the class membership of only a subset of subjects (survivors) were subsequently labeled. In early analyses of the dataset, challenges with separation of points and asymmetric confidence intervals were encountered. This dissertation extends the previous analyses by characterizing the model in terms of a mixture of penalized (Firth) logistic regressions and developing methods for constructing profile likelihood-based confidence and inverse intervals, and confidence bands in the context of such a model. The proposed methods are applied to the motivating dataset and another related dataset, resulting in improved inference on model parameters. Additionally, a simulation experiment is carried out to further illustrate the benefits of the proposed methods and to begin to explore better designs for future studies. The penalized model is shown to be less biased than the traditional model and profile likelihood-based intervals are shown to have better coverage probability than Wald-type intervals. Some limitations, extensions, and alternatives to the proposed methods are discussed.

A probability distribution is a statistical function that describes the probability of possible outcomes in an experiment or occurrence. There are many different probability distributions that give the probability of an event happening, given some sample size n. An important question in statistics is to determine the distribution of the sum of independent random variables when the sample size n is fixed. For example, it is known that the sum of n independent Bernoulli random variables with success probability p is a Binomial distribution with parameters n and p: However, this is not true when the sample size is not fixed but a random variable. The goal of this thesis is to determine the distribution of the sum of independent random variables when the sample size is randomly distributed as a Poisson distribution. We will also discuss the mean and the variance of this unconditional distribution.

Call number: LD2668 .T4 1978 A65 / Master of Science

published_or_final_version / abstract / toc / Computer Science and Information Systems / Master / Master of Philosophy

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

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