Compute $E(X\mid X+Y)$ if $(X,Y)$ is centered normal with known covariance matrix

Question

The random variable $(X,Y)$ has a two dimensional normal distribution with mean $(0,0)$ and covariance matrix $\begin{pmatrix} 4&2 \\ 2&2 \end{pmatrix}$. Find $E(X\mid X+Y)$.

I am completely lost with this question.

Answer 1

Let's pose $Z = X+Y$. The random variable $(X,Z)$ has a two dimensional normal distribution with mean $$(E(X),E(Z)) ) (E(X), E(X)+E(Y)) = (0,0)$$ and covariance matrix $$\pmatrix{E(X^2) & E(XZ)\\E(ZX) & E(Z^2)} = \pmatrix{E(X^2) & E(X^2)+E(XY) \\E(X^2) + E(YX) & E(X^2)+E(Y^2)+E(2XY)} = \\ \pmatrix{4 & 4+2 \\4 + 2 & 4 + 2 + 2\cdot2} = \pmatrix{4 & 6\\6 & 10}.$$

The correlation coefficient of $X$ and $Z$ is:

$$\rho = \frac{E(XZ)}{\sqrt{E(X^2)E(Z^2)}} = \frac{6}{\sqrt{4 \cdot 10}} = 3\frac{\sqrt{10}}{10}.$$

The conditional expected value is:

$$E(X\mid Z) = \mu_X + \rho \sqrt{\frac{E(X^2)}{E(Z^2)}}(Z-\mu_Z) = \\ = 0 + 3\frac{\sqrt{10}}{10}\sqrt{\frac{4}{10}}(Z-0) = \frac{3}{5}Z.$$

For the last step, take a look here.

Answer 2

Hint $$Z=X + Y \sim N(\mu_X + \mu_Y,\; \sigma_X^2 + \sigma_Y^2 + 2\sigma_{X,Y})$$ $$Z=X + Y \sim N(0\, ,\; 10)$$ Moreover $$f_{X\mid Z}(x,z)=\frac{f_{X,Z}(x,z)}{f_Z(z)}$$ $$\operatorname{Cov}(Z,X)=\operatorname{Cov}(X+Y,X)=\operatorname{Var}(X) + \operatorname{Cov}(X,Y)=4+2=6$$