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.
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We introduce a simple and powerful procedure-the observer method-in order to obtain a reliable method of numerical integration over an arbitrary long interval of time for systems of ordinary differential equations having first integrals. This aim is achieved by a modification of the original system such that the level manifold of the first integrals becomes a local attractor. We provide a theoretical justification of this procedure. We report many tests and examples dealing with a large spectrum of systems with different dynamical behaviour. The comparison with standard and symplectic methods of integration is also provided.
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