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

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.

## closed as off-topic by Did, Henry Swanson, Claude Leibovici, Watson, quid♦Sep 8 '16 at 9:48

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Henry Swanson, Claude Leibovici, Watson, quid
If this question can be reworded to fit the rules in the help center, please edit the question.

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.

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 $$f_{X\mid Z}(x,z)=\frac{f_{X,Z}(x,z)}{f_Z(z)}$$ $$\operatorname{Cov}(Z,X)=\operatorname{Cov}(X+Y,X)=\operatorname{Var}(X) + \operatorname{Cov}(X,Y)=4+2=6$$