This study deals with the problem of moral development discontinuity prevalent in today's multicultural societies. Black adolescents are confronted by many obstacles in their situatedness in the home, school and society. It has been revealed that the black adolescent in the multicultural situation is exploited, dehumanised and exposed to impersonal situations when he should be offered moral guidance and challenging moral dilemmas in order to develop his moral sense, however he is consequently reduced to an object that is tossed to and fro by his fellow human beings. The black adolescent, it has been shown, needs his fellow human beings, as transmitters of moral values to help him to achieve a moral-self. This study examines the three multicultural situations, the home, the school and the society and shows that morals are not inherited but acquired through mutual contact. The acquisition of morals manifests itself under conditions characterised by respect, modelling, imitation, indoctrination, reward and punishment, conformity, loyalty, communication, exemplification, socialisation, experience and learning as determined by the home, school and society. It was also found that in their acquisition of morals in a multicultural society, black adolescents experience confusion brought about by the cultural differences of their society. \\!hat they previously regarded as the right thing to do in their cultural background receives negative responses in the multicultural situation. It was further found that black adolescents in a multicultural situation are not provided with sufficient opportunities to participate meaningfully in moulding their new moral environment. The multicultural environment is cold and unfriendly, as a result black adolescents are barred from expanding and anchoring themselves in their new situation to face the challenges confronting them with confidence. The empirical research revealed that in the home parents are too busy with their professional upgrading and the positions they hold at work to bother about the moral upbringing of their children. In school teachers emphasise scholastic achievement above moral development. The society does not provide black adolescents with moral role models to imitate. Society has become to technocratic, with devices such as the TV, radio, Internet and video games, to guide black adolescents in their moral intemalisation. / Psychology of Education / D. Ed. (Psychology of Education)

In this thesis, we present some new results in distribution theory for both discrete and continuous random variables, together with their motivating applications. We start with some results about the Multivariate Gaussian Distribution and its characterization as a maximizer of the Strichartz Estimates. Then, we present some characterizations of discrete and continuous distributions through ideas coming from optimal transportation. After this, we pass to the Simpson's Paradox and see that it is ubiquitous and it appears in Quantum Mechanics as well. We conclude with a group of results about discrete and continuous distributions invariant under symmetries, in particular invariant under the groups $A_1$, an elliptical version of $O(n)$ and $\mathbb{T}^n$. As mentioned, all the results proved in this thesis are motivated by their applications in different research areas. The applications will be thoroughly discussed. We have tried to keep each chapter self-contained and recalled results from other chapters when needed. The following is a more precise summary of the results discussed in each chapter. In chapter \ref{chapter 2}, we discuss a variational characterization of the Multivariate Normal distribution (MVN) as a maximizer of the Strichartz Estimates. Strichartz Estimates appear as a fundamental tool in the proof of wellposedness results for dispersive PDEs. With respect to the characterization of the MVN distribution as a maximizer of the entropy functional, the characterization as a maximizer of the Strichartz Estimate does not require the constraint of fixed variance. In this chapter, we compute the precise optimal constant for the whole range of Strichartz admissible exponents, discuss the connection of this problem to Restriction Theorems in Fourier analysis and give some statistical properties of the family of Gaussian Distributions which maximize the Strichartz estimates, such as Fisher Information, Index of Dispersion and Stochastic Ordering. We conclude this chapter presenting an optimization algorithm to compute numerically the maximizers. Chapter \ref{chapter 3} is devoted to the characterization of distributions by means of techniques from Optimal Transportation and the Monge-Amp\`{e}re equation. We give emphasis to methods to do statistical inference for distributions that do not possess good regularity, decay or integrability properties. For example, distributions which do not admit a finite expected value, such as the Cauchy distribution. The main tool used here is a modified version of the characteristic function (a particular case of the Fourier Transform). An important motivation to develop these tools come from Big Data analysis and in particular the Consensus Monte Carlo Algorithm. In chapter \ref{chapter 4}, we study the \emph{Simpson's Paradox}. The \emph{Simpson's Paradox} is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, the statistical implications are deep. Basic examples appear in arithmetic, geometry, linear algebra, statistics, game theory, sociology (e.g. gender bias in the graduate school admission process) and so on and so forth. In our new results, we prove the occurrence of the \emph{Simpson's Paradox} in Quantum Mechanics. In particular, we prove that the \emph{Simpson's Paradox} occurs for solutions of the \emph{Quantum Harmonic Oscillator} both in the stationary case and in the non-stationary case. We prove that the phenomenon is not isolated and that it appears (asymptotically) in the context of the \emph{Nonlinear Schr\"{o}dinger Equation} as well. The likelihood of the \emph{Simpson's Paradox} in Quantum Mechanics and the physical implications are also discussed. Chapter \ref{chapter 5} contains some new results about distributions with symmetries. We first discuss a result on symmetric order statistics. We prove that the symmetry of any of the order statistics is equivalent to the symmetry of the underlying distribution. Then, we characterize elliptical distributions through group invariance and give some properties. Finally, we study geometric probability distributions on the torus with applications to molecular biology. In particular, we introduce a new family of distributions generated through stereographic projection, give several properties of them and compare them with the Von-Mises distribution and its multivariate extensions. / Thesis / Doctor of Philosophy (PhD)