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The analysis of the product of two normally distributed variables does not seem to follow any known distribution. Fortunately, the moment-generating function is available and we can calculate the s...

[Abstract]: The distribution of the product of two normally distributed random variables has been an open problem from the early years in the XXth century. First approaches tried to determinate the mathematical and statistical properties of the distribution of such a product using different types of functions. Recently, an improvement in computational techniques has performed new approaches for calculating related integrals by using numerical integration. Another approach is to adopt any other distribution to approximate the probability density function of this product. The skew-normal distribution is a generalization of the normal distribution which considers skewness making it flexible. In this work, we approximate the distribution of the product of two normally distributed random variables using a type of skew-normal distribution. The influence of the parameters of the two normal distributions on the approximation is explored. When one of the normally distributed variables has an inverse coefficient of variation greater than one, our approximation performs better than when both normally distributed variables have inverse coefficients of variation less than one. A graphical analysis visually shows the superiority of our approach in relation to other approaches proposed in the literature on the topic. Universidad de Lisboa; UIDB/00006/2020 Agencia Nacional de Investigación y Desarrollo del gobierno de Chile; 1200525

The estimation of variance components and estimable vectors is carried out for nonorthogonal mixed models. To do that we use a decomposition of the design space in several orthogonal blocks. We assume the random vectors to have null mean vectors and null cross covariance matrices.

This paper presents a study of the evolution the study of the product of two normally distributed variables. From first approaches in the middle of the 20th Century until the most recent ones from this decade. Nowadays, existence of an unique formula for the product was not proved. Partial results using different approaches: Bessel Functions, Numerical Integration, Pearson Type Function, …

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In this work the probability density function (PDF) for the product of two uniformly distributed random variables is explored under the implementation of a new procedure in R language. Based on the Rohatgi theorem for the theoretical form of the product, different possibilities for the range of values of the limits of both distributions are considered. As an application, the management of a (Q,r) inventory model with the presence of lead-time and uniform demand forecasts is considered. Solution to this model looks up to minimize the total costs through the variables Q (reorder quantity) and r (the reorder point), and not always exists an analytical solution of the problem. We show a graphical procedure for the simulation results and a more exactly analytical solution. Implementation in R is straightforward. info:eu-repo/semantics/publishedVersion

Kurtosis (κ) is any measure of the “peakedness” of a distribution of a real-valued random variable. We study the evolution of the Kurtosis for the product of two normally distributed variables. Product of two normal variables is a very common problem for some areas of study, like, physics, economics, psychology, … Normal variables have a constant value for kurtosis (κ = 3), independently of the value of the two parameters: mean and variance. In fact, the excess kurtosis is defined as κ− 3 and the Normal Distribution Kurtosis is zero. The product of two normally distributed variables is a function of the parameters of the two variables and the correlation between then, and the range for kurtosis is in [0, 6] for independent variables and in [0, 12] when correlation between then is allowed.

For a Peirce algebra P, lattices CongP of all heterogenous Peirce con- gruences and IdeP of all heterogenous Peirce ideals are presented. The notions of kernel of a Peirce congruence and the congruence induced by a Peirce ideal are in- troduced to describe an isomorphism between CongP and IdeP. This isomorphism leads us to conclude that the class of the Peirce algebras is ideal determined. Oppo- sed to Boolean modules case, each part of a Peirce ideal I = (I1;I2) determines the other one. A similar result is valid to Peirce congruences. A characterization of the simple Peirce algebras is presented coinciding to that given by Brink, Britz and Schmidt in a homogeneous approach.

Definitions for heterogeneous congruences and heterogeneous ideals on a Boolean module are given and the respective lattices and are presented. A characterization of the simple bijective Boolean modules is achieved differing from that given by Brink in a homogeneous approach. We construct the smallest and the greatest modular congruence having the same Boolean part. The same is established for modular ideals. The notions of kernel of a modular congruence and the congruence induced by a modular ideal are introduced to describe an isomorphism between and . This isomorphism leads us to conclude that the class of the Boolean module is ideal determined.

A characterization of the subdirectly irreducible separable dynamic algebras is presented. The notions develo- ped for this study were also suitable to describe the previously found class of simple separable dynamic algebras (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The distribution of the ratio of two normal random variables X and Y was studied from [1] (the density function) and [2] (the distribution function). The shape of its density function can be uni-modal, bimodal, symmetric, asymmetric, following several type of distributions, like Dirac Distribution, Normal Distribution, Cauchy Distribution or Recinormal Distribution.In this paper we study a different approximation for this distribution Z = X /Y , as a function of four parameters: ratio of the means of the two normal variables, ratio of the standard deviations of the two normal variables, the variation coefficient of the normal variable Y , and the correlation between the two variables. A formula for the Distribution function and the density function of Z is given. In addition, using graphical procedures we established singularity points for the parameters where the approximation given for Z has a non normal shape.

The main purpose of this work is to introduce the class of the monadic dynamic algebras (dynamic algebras with one quantifier). Similarly to a theorem of Kozen we establish that every separable monadic dynamic algebra is isomorphic to a monadic (possibly non-standard) Kripke structure. We also classify the simple (monadic) dynamic algebras. Moreover, in the dynamic duality theory, we analyze the conditions under which a hemimorphism of a dynamic algebra into itself defines a quantifier. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The assessment of the cardiovascular system condition based on multiple parameters allows a more precise and accurate diagnosis of the heart and arterial tree condition. For this reason, the interest in non-invasive devices has presently increased in importance. In this work, an optical probe was tested in order to validate this technology for measuring multiple parameters such as Pulse Wave Velocity (PWV) or Augmentation Index (AIx), amongst others. The PWV measured by the optical probe was previously compared with the values obtained with the gold-standard system. Another analysis was performed in 131 young subjects to establish carotid PWV reference values as well as other hemodynamic parameters and to find correlations between these and the population characteristics. The results allowed us to conclude that this new technique is a reliable method to determine these parameters. The range of the obtained values for local PWV are in agreement with the values obtained by other studies, and significant correlations with age and smoking status were found. The AIx varied between −6.15 % and 11.46 % and exhibit a negative correlation with heart rate, and dP/dtmax shows a significant decrease with age.

In our work we present an application of some biometrical methods useful in genotype stability evaluation, namely AMMI model, Joint Regression Analysis (JRA) and multiple comparison tests. A genotype stability analysis of oat (Avena Sativa L.) grain yield was carried out using data of the Portuguese Plant Breeding Board, sample of the 22 different genotypes during the years 2002, 2003 and 2004 in six locations. In Ferreira et al. (2006) the authors state the relevance of the regression models and of the Additive Main Effects and Multiplicative Interactions (AMMI) model, to study and to estimate phenotypic stability effects. As computational techniques we use the Zigzag algorithm to estimate the regression coefficients and the agricolae-package available in R software for AMMI model analysis.

The lattice CongD of all dynamic congruences on a given dynamic algebra D is presented. Whenever D is separable with zero we define dynamic ideal on D, given rise to the lattice IdeD. The notions of kernel of a dynamic congruence and the congruence generated by a dynamic ideal are introduced to describe a Galois connection between CongD and IdeD. We study conditions under which a dynamic congruence is determined by its kernel.

Our research is centred on the stochastic structure of matched open populations, subjected to periodical reclassifications. These populations are divided into sub-populations. In our application we considered two populations of customers of a bank: with and without account manager. Two or more of such population are matched when there is a 1-1 correspondence between their sub-populations and the elements of one of them can go to another, if and only if the same occurs with elements from the corresponding sub-populations of the other. So we have inputs 104 C. Rocadas, T.A. Oliveira and J.T. Mexia and outputs of elements in the population and along with several subpopulations in which the elements can be placed. It is thus natural to use Markov chains to model these populations. Besides this study connected with Markov chains we show how to carry out Analysis of Variance – like analysis of entries and departures to and from de populations of customers. Our purpose is to study the flows in and out of customers in classes for the two populations and to make research on the influence of the factors year, class and region. We used the Likelihood ratio tests for the hypotheses formulated on the basis of these factors. In our work we verified that major hypotheses were all rejected. This raises the question of what are the effects and interactions truly relevant. Looking for an answer to this problem, we present the first partition to a change in the log Likelihood. This partition is very similar to the analysis of variance for the crossing of the factors that allowed us to use algebraic established results, see Fonseca et al. (2003, 2006), for models with balanced cross.

In this paper, we study the existence and uniqueness of the solution for Markov renewal equation (MRE) of a semi-Markov process with countable state space. This method and its proof are based on an iterative scheme. A numerical solution is also given as well as a case study on system reliability assessment.