Let X,Y,Z $\stackrel{i.i.d}{\sim}$ N($0$,$1$). I am supposed to find the E($2$X+$3$Y | X+$3$Y-Z =$4$)
I Tried to solve the problem by considering A= $2X+3Y$ ~ $N(5,5)$ and B=X+$3$Y-Z ~ $N(3,5)$ but I was confused when I was trying to find the joint pdf of A and B.
Kindly help as this is my first time in solving such type of problems.
there are some errors:
$$A=2X+3Y\sim N(0;13)$$
$$B=X+3Y-Z\sim N(0;11)$$
Now you can derive how $A$ and $B$ are correlated and using the properties of jointly bivariate Gaussian you shoudl solve your problem finding
$$\mathbb{E}[A|B=4]=4$$
Furhter details:
To calculate $E(AB)$ simply use the definition
$$\mathbb{E}[(2X+3Y)(X+3Y-Z)]=\mathbb{E}[\text{expand}]=\dots=11$$