Suppose we have two independent random variables $X$ and $Y$, with expected values and standard deviations of $(\mu_X,\sigma_X)$ and $(\mu_Y,\sigma_Y)$, respectively. Can we say anything about the expected value and standard deviation of $|X-Y|$?
If it had been $X-Y$, the answer would be $(\mu_X-\mu_Y, \sqrt{\sigma_X^2+\sigma_Y^2})$. However, I think such a straightforward approach is not possible because $P(X>Y)$ is not independent from $P(X,Y)$.
Best Answer
Suppose we know the mean $\mu$ of $|X-Y|$. This is enough to determine the variance, since it is $E(|X-Y|^2)-\mu^2$. For note that $$E(|X-Y|^2)=E(X^2)-2E(XY)+E(Y^2),$$ and each of the terms above can be calculated from the mean and variance of $X$ and $Y$.
Unfortunately, $\mu$ is not determined by the individual means and variances of $X$ and $Y$. For example, let $X$ and $Y$ take on values $-1$ and $1$ each with probability $1/2$. Then $X$ and $Y$ have mean $0$ and variance $1$, and $E(|X-Y|)=1$. But if we use standard normals for $X$ and $Y$, the mean of $|X-Y|$ is different (I believe it is $\frac{2}{\sqrt{\pi}}$).
This does not fully answer the question, since it only says we cannot determine $\mu$ from the given information. There will undoubtedly be useful inequalities.