Covariance of absolute values of random vaiables

Question

How would I go about calculating $\operatorname{cov}(|X|,|Y|)$, if I know $f_{X,Y}(x,y)$ and $\operatorname{cov}(X,Y)$ ?

Answer 1

If you know the joint density function $f_{X,Y}(x,y)$ of $X$ and $Y$, then $\text{Cov}(|X|,|Y|)$ is dealt with in the usual way. For $$E(|X|)=\iint_{\mathbb{R}^2} |x| f_{X,Y}\,dx\,dy,$$ with a similar expression for $E(|Y|)$. And $$E(|XY|)=\iint_{\mathbb{R}^2} |xy| f_{X,Y}\,dx\,dy.$$ We can now use $\text{Cov}(U,V)=E(UV)-E(U)E(V)$ to finish.