On the distribution of the sum of dependent standard normally distributed random variables using copulas
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The distribution function of the sum Z of two standard normally distributed random variables X and Y is computed with the concept of copulas to model the dependency between X and Y. By using implicit copulas such as the Gauss- or t-copula as well as Archimedean Copulas such as the Clayton-, Gumbel- or Frank-copula, a wide variety of different dependencies can be covered. For each of these copulas an analytical closed form expression for the corresponding joint probability density function f_X,Y is derived. We apply a numerical approximation algorithm in Matlab to evaluate the resulting double integral for the cumulative distribution function F_Z. Our results demonstrate, that there are significant differencies amongst the various copulas concerning F_Z. This is particularly true for the higher quantiles (e.g. 0.95, 0.99), where deviations of more than 10% have been noticed.
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