I have a problem which comes down to the following ratio of weighted Gaussian random variables $u_i$,
$$\mathbf{y} = \frac{\sum_{i=1}^N \mathbf{x}_i u_i}{\sum_{i=1}^N u_i},$$
where $\mathbf{x_i} \in \mathbb{R}^D$ are vectors and $u_i$ are i.i.d. Gaussian.
I have been trying to figure out the distribution of $\mathbf{y}$, and even for the special case $N=2$
$$\mathbf{y} = \frac{\mathbf{x}_1 u_1 + \mathbf{x}_2 u_2}{u_1 + u_2},$$
I am not quite sure how to approach it. The challenge is that the numerator and denominator are not independent.
Any pointers are very much appreciated. Thanks a lot!