Suppose that X and Y are two random variable jointly Gaussian with mean
$\mu_X$ and $\mu_Y$, variance $\sigma^2_X$ and $\sigma^2_Y$
respectively, and covariance $C$.
a) Determine the conditional probability density of $X + Y$ given $Y = y$;
b) Find $E [X^2 + Y^2 | X = x, Y = y]$;
c) Find $E [X | Y = y]$ and $Var[X | Y = y]$;
Here is my attempt:
a)$P(X+Y|Y=y) = \frac{P(X)\cap P(Y=y)}{P(Y=y)} = \frac{P(X,y)}{P_Y(y)}$
with $p_Y(y) > 0$
b)$E[X^{2} + Y^{2}|X=x,Y=y] = E[x^2 + y^2] = E[x^2] + E[y^2]$
c.1)$E[X|Y=y] = \int_R xf_{X|Y}(x|y)\mathrm{d}x = \int_R xf_{X}(x)\mathrm{d}x = E(X)$
c.2)$Var[X|Y=y]$
Generally I can say that:
$Var[X|Y] = E((X-E(X|Y))^{2}|Y) \Rightarrow Var[X|Y=y] = E((X-E(X|Y=y))^{2}|Y=y)$
$= \int_x(x-E(X|Y=y))^{2}P_{X|Y}(x)$ but here I'm stuck
Thanks for helping me!
Best Answer
a) You want to find a probability density function, so use symbols like these:
$\qquad\begin{align}f_{X+Y\mid Y}(z\mid y) &= \dfrac{f_{X,Y}(z-y,y)}{f_Y(y)} \end{align}$
As you did later, so why not use them now?
Also remember that $\langle X,Y\rangle\sim{\large\mathcal N}\left(\begin{bmatrix}\mu_X\\\mu_Y\end{bmatrix},\begin{bmatrix}\sigma_X^2&C\\C&\sigma_Y^2\end{bmatrix}\right)$ so you should know of these pdf.
b) The expectation of scalar values are just the values themselves.
$\qquad\mathsf E(X^2+Y^2\mid X=x, Y=y)=x^2+y^2$
c) No, $f_{X\mid Y}(x\mid y)$ does not equal $f_{X}(x)$ when the covariance ($C$) is not zero. Find it similarly to part $a$.
$\qquad\mathsf E(X\mid Y=y)=\int_\Bbb R \frac{x~f_{X,Y}(x,y)}{f_Y(y)}\,\mathrm d x\\\quad\mathsf {Var}(X\mid Y=y) = \int_\Bbb R(x-\mathsf E(X\mid Y=y))^2 \frac{f_{X,Y}(x,y)}{f_Y(y)}\,\mathrm d x$
But also see the bivariate case for Multivariate Normal Conditional Distribution on Wikipedia.