Minimum mean square estimation with 3 random variable

I am trying to find the solution for part "a", I have try finding the expectaion

$E[x|y=y] = \int x f(x|y=y) dx = \int x \frac{f(y=y|x=x)f(x))}{f(y=y)}$.

With y|x ~ Normal (2x,1). Yet it gets me no where, as this turn out to be a quite complicated differentiation. I wonder if I did anything wrong or is there a different approach that I do not know about

Solution 1:

I am trying to find $\mathbb{E}[X|Y=y]$

Observe that $(X,Y)$ are jointly gaussian

$$(X,Y)\sim N\left(0;0;1;5;\frac{2}{\sqrt{5}}\right)$$

thus the joint density can be factorized as follows

$$f_{XY}(x,y)=f_Y(y)\cdot f_{X|Y}(x|y)$$

where

$$(X|Y=y)\sim N\left(\mu_X+\rho\frac{\sigma_X}{\sigma_Y}(y-\mu_Y);\sigma_X^2(1-\rho^2) \right)$$

that, in your example is

$$(X|Y=y)\sim N(0.4y;0.2)$$

Thus

$$\mathbb{E}[X|Y=y]=0.4y$$