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In this paper, we first present the basic principles of set theory and combinatorial analysis which are the most useful tools in computing probabilities. Then, we show some important properties derived from axioms of probability. Conditional probabilities come into play not only when some partial information is available, but also as a tool to compute probabilities more easily, even when partial information is unavailable. Then, the concept of random variable and its some related properties are introduced. For univariate random variables, we introduce the basic properties of some common discrete and continuous distributions. The important properties of jointly distributed random variables are also considered. Some inequalities, the law of large numbers and the central limit theorem are discussed. Finally, we introduce additional topics the Poisson process.

The Birnbaum-Saunders (BS) distribution is a positively skewed distribution and is a popular model for analyzing lifetime data. In this thesis, we first develop an improved method of estimation for the BS distribution and the corresponding inference. Compared to the maximum likelihood estimators (MLEs) and the modified moment estimators (MMEs), the proposed method results in estimators with smaller bias, but having the same mean squared errors (MSEs) as these two estimators. Next, the existence and uniqueness of the MLEs of the parameters of BS distribution are discussed based on Type-I, Type-II and hybrid censored samples. In the case of five-parameter bivariate Birnbaum-Saunders (BVBS) distribution, we use the distributional relationship between the bivariate normal and BVBS distributions to propose a simple and efficient method of estimation based on Type-II censored samples. Regression analysis is commonly used in the analysis of life-test data when some covariates are involved. For this reason, we consider the regression problem based on BS and BVBS distributions and develop the associated inferential methods. One may generalize the BS distribution by using Laplace kernel in place of the normal kernel, referred to as the Laplace BS (LBS) distribution, and it is one of the generalized Birnbaum-Saunders (GBS) distributions. Since the LBS distribution has a close relationship with the Laplace distribution, it becomes necessary to first carry out a detailed study of inference for the Laplace distribution before studying the LBS distribution. Several inferential results have been developed in the literature for the Laplace distribution based on complete samples. However, research on Type-II censored samples is somewhat scarce and in fact there is no work on Type-I censoring. For this reason, we first start with MLEs of the location and scale parameters of Laplace distribution based on Type-II and Type-I censored samples. In the case of Type-II censoring, we derive the exact joint and marginal moment generating functions (MGF) of the MLEs. Then, using these expressions, we derive the exact conditional marginal and joint density functions of the MLEs and utilize them to develop exact confidence intervals (CIs) for some life parameters of interest. In the case of Type-I censoring, we first derive explicit expressions for the MLEs of the parameters, and then derive the exact conditional joint and marginal MGFs and use them to derive the exact conditional marginal and joint density functions of the MLEs. These densities are used in turn to develop marginal and joint CIs for some quantities of interest. Finally, we consider the LBS distribution and formally show the different kinds of shapes of the probability density function (PDF) and the hazard function. We then derive the MLEs of the parameters and prove that they always exist and are unique. Next, we propose the MMEs, which can be used as initial values in the numerical computation of the MLEs. We also discuss the interval estimation of parameters. / Thesis / Doctor of Science (PhD)

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A simulation matrix population model of a small population of wrens (Troglodytes troglodytes) is presented. The field work methods used to obtain and analyse the demographic rates are provided. This includes a description of the use of miniature radio tags to track juvenile (post-fledging) survival and dispersal, and capture mark recapture analysis of an eight year dataset to estimate adult survival rates, taking into account environmental variation and density dependence. Age related reproductive rates were obtained from detailed nest surveys. Using these demographic rates (means and variances), and information on density dependence in survival and breeding, a simulation matrix model was developed using Matlab (The MathWorks, Inc.). The operation of this model and its outputs are explained in detail, with particular reference to the methods employed to incorporate both density dependent survival and reproduction and environmental and demographic stochasticity. This model is then used to illustrate how, under plausible conditions of density dependence and stochasticity, large discrepancies are obtained between the deterministic, density independent elasticities of the population growth rate (λ) and the stochastic, density dependent elasticities of the equilibrium population size, extinction probability and invasion exponent. Since the elasticities of λ are often used to guide the management of endangered species, these results are particularly relevant to workers in the field of rare species conservation. While the importance of including environmental variation in the form of stochastic population simulations seems to now be generally accepted, the role of density dependent population regulation is still infrequently considered. Since one of the most common causes of population decline is habitat destruction, leading to an increase in population density within the remaining areas of habitat, this omission may rarely be justified. It is recommended that when elasticity analysis is conducted as part of species conservation efforts, both density dependence and stochasticity are included. Failure to do so may result in the misguided management of endangered species.

Ce manuscrit aborde le problème de l'atomisation de films liquides à travers diverses expériences portant sur leur formation, déstabilisation et fragmentation. Une première partie identifie et décrit deux mécanismes de désintégration à l'œuvre lors de la rétractation d'un film de savon. Lorsque le film se détache sur une frontière, le bord de Plateau est entrainé et son accélération explique la déstabilisation. Lorsque le perçage a lieu au centre, le battement du film résulte de l'interaction avec l'atmosphère au repos, ce qui conduit à sa fragmentation. Une seconde étude explique le plissement singulier de certaines cloches liquides par un ressaut hydraulique à interfaces libres assurant la transition d'un écoulement supercritique vis-à-vis des ondes capillaires à un écoulement subcritique. La courbure marquée associée à cette transition conduit à une accélération centripète du liquide susceptible de déstabiliser la cloche en aval. La formation des embruns par l'intermédiaire de l'éclatement des bulles de surface est abordée dans une troisième partie. La géométrie des bulles, le mécanisme de drainage de leurs films et les événements de perçage sont étudiés afin de rendre compte de l'épaisseur de la bulle au début de l'éclatement. La dynamique ultérieure de fragmentation est détaillée conduisant à une caractérisation du spray produit. Une dernière partie étudie l'atomisation dite effervescente sur une expérience modèle : une nappe liquide plane ensemencée en fines bulles d'air favorisant la nucléation de trous. Sur la base de mesures des taux de nucléation et croissance des trous, un modèle statistique du réseau de ligaments liquides formés par la réunion des bourrelets ceinturant les trous permet un description du spray issu de la fragmentation du réseau. / This thesis tackles the problem of liquid films atomization through various experimental configurations dealing with their formation, destabilization and fragmentation. In a first part, two mechanisms of receding soap film disintegration are identified and described. When the film is released from a border, the acceleration sustained by the Plateau border explains the destabilization. When puncture occurs in the center, the film flapping is promoted by the interaction with the surrounding atmosphere which leads to its fragmentation. In a second study, the singular folding of liquid bells is explained by an original free surfaces hydraulic jump insuring transition from a subcritical to a supercritical flow with respect to capillary waves. The strong curvatures associated with this transition, also implying strong centripetal acceleration of the flowing liquid, may induce the downstream sheet destabilization. The question of marine aerosols formation via surface bubble bursting is investigated in a third part. Bubbles geometry, their film drainage mechanism and puncture events are studied in order to account for the film thickness at the onset of bursting. The subsequent bursting dynamics is detailed to characterize the spray production. In a last part, effervescent atomization is studied on a model experimental set up: a flat liquid sheet seeded with small air bubbles acting as heterogeneous hole nuclei. On the basis of holes nucleation and growth measurements, a statistical model of the liquid ligament red resulting from the holes rims combination offers a description of the spray generated by the red fragmentation.

The theme of this study is spatial mismatch, a concept that gave rise to an ever-expanding body of research concerned with how and why residential and employment distributions have shifted within cities and across metropolitan areas. The concept grew out of John F. Kain's research on how racial discrimination and segregation affects the spatial patterns of people/subgroups and jobs in the postwar American urban environment. Specifically, "Housing Segregation" posits that housing-market discrimination is at the root of increased unemployment among inner-city, nonwhite workers; concurrently, the pace and volume of decentralization (of residents and employment) from central-cities reinforces low-income, overwhelmingly African-American isolation and immobility. This study contributes to the New Orleans literature by providing a pre- and post-Katrina snapshot of spatial mismatch. The analysis addresses research questions aimed at gauging the extent to which mismatch and job-isolation have changed for poor workers in the New Orleans metro area since Hurricane Katrina.

The supLM test for structural change is embedded into a permutation test framework for a simple location model. The resulting conditional permutation distribution is compared to the usual (unconditional) asymptotic distribution, showing that the power of the test can be clearly improved in small samples. Furthermore, generalizations are discussed for binary and multivariate dependent variables as well as model-based permutation testing for structural change. The procedures suggested are illustrated using both artificial and real-world data (number of youth homicides, employment discrimination data, structural-change publications, and stock returns). / Series: Research Report Series / Department of Statistics and Mathematics

A dissertation submitted to the Faculty of Engineering. University of the Witwatersrand. Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering. D.tT~.awart •• w1t~4i.tinotion on , Dec ••ber I,e7 Johannesburg. ItN7 / Accurate forecastinu of system Maximum Demand (MDl is vital ~o Esk011l. Under-estimating the MD could re-ul: in a generation capacity shortage. with devastating consequences for the economy. Similarly. a high MD forecast would result in overcapacity. with expensive generating plant standing idle The traditional method of MD forecasting in Eskorn has become unreliable due to a changing relationship between forecast energy sales and expected maximum demand. The reasons for the changing system demand profile were isolated and analysed. Alternative MD forecasting techniques are evaluated and end-use hourly aggregation modets were ;d'~ntifieJ as a method suitable for Eskorn, An experimental demand profile forecasting moue! was developed, using data from a previous project. The model was tested and proved able to cope with the structural changes in the system demand profile. This resulted in the adoption of this technique by Eskorn and approval for the devclopn-ent of a f\.JJ1 scale de manu profile forecasting model.