  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.
