What is the distribution of a function of Gaussian independent random variables

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 $f$?

• The trouble with this approach (and the reason why it fails) is that the transform $(X_1,X_2)\mapsto (Z_1,Z_2)=(X_1/X_2,X_2)$ is not linear hence $(Z_1,Z_2)$ is not Gaussian. – Did Jan 6 '16 at 17:51
• what would be the the alternative? – Lost in Jan 8 '16 at 15:20
• To use the arch classical change of variables formula based on the Jacobian of the transformation. – Did Jan 8 '16 at 16:32