Suppose that Y~$N\begin{pmatrix} 1\\ 2\end{pmatrix},\begin{pmatrix}2 & 1\\ 1 & 2 \end{pmatrix}$. How can I find the conditional PDF of $Y_1$ given that $Y_1+Y_2=3$?? I am given a hint to firstly obtain the joint density of $Y_1$ and $Y_1+Y_2$ and then use the property of the conditional pdf that $f(x|z)=f(x,z)/f(z)$.
So far, I have tried to find the joint density of $Y_1$ and $Y_1+Y_2$. so I concluded to $\begin{pmatrix} Y_{1} \\ Y_{1}+Y_{2} \end{pmatrix} $~$N \begin{pmatrix} 1\\ 3 \end{pmatrix} , \begin{pmatrix} 3 & 0\\ 0 & 6\end{pmatrix}$ but I feel that this is not correct.
Any help would be appreicated!!