# How to find expectation of absolute values just from the data given below?

Suppose that $X$ and $Y$ are random variables such that

$$E(X + Y ) = E(X - Y ) = 0 ;$$

$$\operatorname{Var}(X + Y ) = 3 ;$$

$$\operatorname{Var}(X - Y ) = 1.$$

(a) Evaluate $\operatorname{Cov}(X,Y)$.

(b) Show that $E|X + Y| \le \sqrt{3}$.

(c) If in addition, it is given that $(X,Y)$ is bivariate normal, calculate $E(|X + Y|^3)$.

Question (a) is solved and I guess question (c) won't be quite difficult since $X+Y$ follows a univariate Normal distribution.

I am stuck with question (b).

• (c) is easier than it might look because the substitution $Z=(X+Y)^2$ converts the expectation into an expression that is a multiple of $\Gamma(2) = \int_0^\infty z\exp(-z)dz.$ For (b), what can you say about $\operatorname{Var}(|X+Y|)$? – whuber Apr 15 '16 at 23:22
• I just solved (b). I used Cauchy-Schwartz Inequality. It didn't occur to me all this while. And yes (c) takes the form of a gamma integral. – user666 Apr 15 '16 at 23:51
• I'd have done b by using the fact that $E(|X+Y|^2)- E(|X+Y|)^2\geq 0$. – Glen_b -Reinstate Monica Apr 16 '16 at 0:13