An approach to distribution of the product of two normal variables

The distribution of product of two normally distributed variables come from the first part of the XX Century. First works about this issue were [1] and [2] showed that under certain conditions the product could be considered as a normally distributed. A more recent approach is [3] that studied approximation to density function of the product using three methods: numerical integration, Monte Carlo simulation and analytical approximation to the result using the normal distribution. They showed as the inverse variation coefficient µ/σ increases, the distribution of the product of two independent normal variables tends towards a normal distribution. Our study is focused in Ware and Lad approaches. The objective was studying which factors have more influence in the presence of normality for the product of two independent normal variables. We have considered two factors: the inverse of the variation coefficient value µ/σ and the combined ratio (product of the two means divided by standard deviation): (µ1µ2(/σ for two normal variables with the same variance. Our results showed that for low values of the inverse of the variation coefficient (less than 1) normal distribution is not a good approximation for the product. Another one, influence of the combined ratio value is less than influence of the inverse of coefficients of variation value.info:eu-repo/semantics/publishedVersio

Skewness into the product of two normally distributed variables and the risk consequences

The analysis of skewness is an essential tool for decision-making since it can be used as an indicator on risk assessment. It is well known that negative skewed distributions lead to negative outcomes, while a positive skewness usually leads to good scenarios and consequently minimizes risks. In this work the impact of skewness on risk analysis will be explored, considering data obtained from the product of two normally distributed variables. In fact, modelling this product using a normal distribution is not a correct approach once skewness in many cases is different from zero. By ignoring this, the researcher will obtain a model understating the risk of highly skewed variables and moreover, for too skewed variables most of common tests in parametric inference cannot be used. In practice, the behaviour of the skewness considering the product of two normal variables is explored as a function of the distributions parameters: mean, variance and inverse of the coefficient variation. Using a measurement error model, the consequences of skewness presence on risk analysis are evaluated by considering several simulations and visualization tools using R software.info:eu-repo/semantics/publishedVersio

Approximating the distribution of the product of two normally distributed random variables

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.info:eu-repo/semantics/publishedVersio

Distorsiones de los sistemas parlamentarios de representación. El caso español

Systems of parliamentary representation are the base of the indirect democracy; where a group of representatives, chosen by means of an electoral process take decisions in the name of the citizens. The systems of traditional representation establish a distribution of the total number of representatives in quotas between different electoral districts, that have to be approximated because of the need to use whole numbers. This fact has given place to several types of dysfunctions in the System, in particular two have been widely analysed in the literature: the level of mal-apportionment and disproportionality. In this work, considering the measures established to quantify these effects, we development an analysis of the real Spanish Congress identifying the provinces prejudiced and benefited by these dysfunctions. At last, a brief study of the power index of the province is developed and an alternative distribution of the seats into the districts is considered.Los sistemas de representación parlamentaria establecen la base de los Sistema denominados de democracia indirecta; donde un grupo de representantes elegidos mediante un proceso electoral toman las decisiones en nombre de los ciudadanos. Los sistemas de representación tradicionales establecen un reparto del número total de representantes en cuotas entre los diferentes distritos electorales, que se deben aproximar debido a la necesidad de utilizar números enteros. Este hecho ha dado lugar a diversos tipos de problemas o disfunciones en el Sistema, en particular dos han sido ampliamente analizados en la literatura: el grado de “mal-apportionment” y el grado de representatividad. En este trabajo a partir de las medidas establecidas para medir dichos efectos, se realiza un análisis de la situación actual en el Congreso de los diputados del parlamento español, identificando las provincias perjudicadas y beneficiadas por estas disfunciones. Por último, se ha hecho un pequeño estudio sobre el índice de poder que presentan las diferentes provincias y una propuesta de reparto alternativa

Distribution function for the ratio of two normal random variables

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 unimodal, 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.info:eu-repo/semantics/publishedVersio

The skewness and kurtosis of the product of two normally distributed random variables

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 statistics of the product distribution: mean, variance, the skewness and kurtosis (excess of kurtosis). In this work, we have considered the role played by the parameters of the two normal distributions’ factors (mean and variance) on the values of the skewness and kurtosis of the product. Ranges of variation are defined for kurtosis and the skewness. The determination of the evolution of the skewness and kurtosis values of the product can be used to establish the normality of the product and how to modelize its distribution. Finally, the Pearson Inequality is proved for the skewness and kurtosis of the product of two normal random variables.info:eu-repo/semantics/publishedVersio

Evaluation of Kurtosis into the product of two normally distributed variables

Kurtosis (k) 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 (k = 3), independently of the value of the two parameters: mean and variance. In fact, the excess kurtosis is defined as k - 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.info:eu-repo/semantics/publishedVersio

Aplicación del método del conjunto activo al equilibrio estático del usuario en redes

El objetivo central de este trabajo es comprobar el comportamiento del método del conjunto activo como medio de asignación óptima en redes de tráfico con presencia de congestión. Diversos estudios proponen la utilización de métodos basados en la idea del conjunto activo, o en la dirección del gradiente a la hora de establecer el nivel de tráfico óptimo en los arcos de una red. En este trabajo se plantea una propuesta similar. Consideramos una red de tráfico donde las funciones de coste de los arcos son del tipo BPR, la demanda de flujo entre los orígenes y destinos está dada y es constante. No existen valoraciones temporales; es, por tanto, una aproximación puramente estática al problema. Tampoco hemos considerado la posible aproximación estocástica. En el óptimo, la red asignará todo el tráfico a los diversos arcos de forma tal que el coste global de funcionamiento de la red será el mínimo; asimismo, el coste de todas las rutas posibles será igual, de modo que ningún usuario estará dispuesto a modificar su ruta de forma individual puesto que no le supondrá una reducción el coste del viaj

Product of two normal variables: an historical review

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 di erent approaches: Bessel Functions, Numerical Integration, Pearson Type Function.info:eu-repo/semantics/publishedVersio