suppose $\mathbf{X_{i}}\in N(\mu_{i},\sigma_{i})$ in order to find distribution of $\mathbf{V}=f(\mathbf{X}_{1},\mathbf{X}_{2})=\frac{\mathbf{X_{1}}}{\mathbf{X_{2}}}$ would it be correct to compute $$Cov(\mathbf{X},\mathbf{Y})=\iint (\mathbf{X_{1}}-\mu_{1})(\mathbf{X_{1}}-\mu_{2})f_{\mathbf{X_{2}},\mathbf{X_{2}}} \:dx_{1}dx_{2}$$
given this and mean for both, the covariance matrix can be defined, Furthermore define a transformation matrix $B$ and constant vector $\bar{a}$ such that $$\bar{Z}=B\bar{X}+\bar{a}$$ there $\mathbf{Z}_{1}=\frac{\mathbf{X_{1}}}{\mathbf{X_{2}}}$ and $\mathbf{Z}_{2}=\mathbf{X}_{2}$
then $$\mathbf{Z}\in N(B\bar{\mu}+\bar{a},B\Sigma B^{T})$$ and finally distribution of $\mathbf{V}=\mathbf{Z}_{1}$
is this really the most efficient way? what can be said about $f$?