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The distribution function of the sum $Z$ of two standard normally distributedrandom variables $X$ and $Y$ is computed with the concept of copulas to modelthe dependency between $X$ and $Y$. By using implicit copulas such as theGauss- 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. Foreach of these copulas an analytical closed form expression for thecorresponding joint probability density function $f_{X,Y}$ is derived. We applya numerical approximation algorithm in Matlab to evaluate the resulting doubleintegral for the cumulative distribution function $F_Z$. Our resultsdemonstrate, that there are significant differencies amongst the variouscopulas 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.

On the distribution of the sum of dependent standard normally distributed random variables using copulas