How to find $E|X+Y|^3$ from related information? Assume that
$$ E(X+Y)=E(X-Y)=0 $$
$$ V(X+Y)=3 $$
$$ V(X-Y)=1 $$


*

*Show that $E|X+Y|\leq\sqrt3$.

*If in addition, it is given that $(X,Y)$ is bivariate normal, calculate $E|X+Y|^3$.
For the 1st part, I used the inequality that
$$ E|X+Y|\leq \sqrt{E(X+Y)^2}=\sqrt3. $$
But I don't know how to approach the 2nd part. One thing that I understood is,
$$ X+Y\sim N(0,\sigma_1^{2}+\sigma_2^{2}+2\sigma_1\sigma_2\rho), $$
where 
$$ V(X)=\sigma_1^{2} $$ 
$$ V(Y)=\sigma_2^{2}. $$
Here we can see that $Var(X)+Var(Y)=2$. How can we find them separately from the given data? Also $E|X+Y|^3$?
 A: Let $Z=X+Y$


*

*We're given $E(Z)=0$, so
$\text{Var}(Z) = E(Z^2) - E(Z)^2 = E(Z^2)$
Now let $U=|Z|$
Use basic properties of variance to relate the variance of $U$ to $E(|Z|)$, from which the needed result follows.

*We have $Z\sim N(0,3)$ and we want to find $E(|Z|^3)$.
It's easy to do it for a standard normal (it should be clear that the standard normal calculation then needs to be scaled by $\sigma^3$ to yield the actual result).
It can be done directly by simply writing the integral for the expectation as twice the integral from $0$ to $\infty$ (by symmetry), and then using the obvious change of variable, reducing to a simple gamma integral you should know by heart (but if you don't know how to do it, try integration by parts).

As for your remaining questions:
Expand $\text{Var}(X+Y)$ and $\text{Var}(X-Y)$,
Looking at both the sum and difference of the two variances will let you separate the covariance from the sum of the variances. However it will not tell you how the sum of the variance splits up.
