$W_{t}$ is Wiener process. I came to a conclusion, that I need to find conditional density. $f_{X|Y} (x, y)$. To do that I need to find $f_{X, Y} (x, y)$ I tried to express both sides somehow, so that they are independent (or divide into a sum, so that both parts are independent) but no luck.
How can I find it? I mean, is there a dumb but working approach here? Or I should keep trying to find some way to express $W_{I}$?
Use the fact that $(W_1+W_2,W_3)\sim N(0, \Sigma)$, where $$ \Sigma=\begin{bmatrix} 5 & 3 \\ 3 & 3 \end{bmatrix}. $$ Then $W_1+W_2\mid W_3=w\sim N(w, 2)$.