Let $ X $ and $Y$ be two random variables with means $\mu_X$ and $\mu_Y$ respectively, as well as variances $\sigma_X$ and $\sigma_Y$ (all of which exist). I am interested in computing the following variance:
$$ Var[sgn(X-Y)]$$
where, of course, sgn denotes the Signum Function.
I am stuck because the closed form of $sgn(X-Y)$ is of the form $(X-Y)/|X-Y|$, at least for $X \ne Y$, and I don't see any straightforward ways of calculating the variance of this quantity. Does anyone know how to go about this?
Thanks in advance.
Edit: We may assume $Cov(X, Y)$ exists.
Best Answer
We have that $$\begin{align} \operatorname E\mathrm{sgn}(X-Y) & =-1\cdot\Pr(X<Y)+0\cdot\Pr(X<Y)+1\cdot\Pr(X>Y) \\ & =\Pr(X>Y)-\Pr(X<Y) \end{align}$$ and $$\begin{align} \operatorname E\mathrm{sgn}^2(X-Y) & =(-1)^2\cdot\Pr(X<Y)+0^2\cdot\Pr(X<Y)+1^2\cdot\Pr(X>Y) \\ & =\Pr(X>Y)+\Pr(X<Y) \end{align}$$ using the law of the unconscious statistician. Hence, $\begin{align} \operatorname{Var}\mathrm{sgn}(X-Y) &=\Pr(X>Y)+\Pr(X<Y)-[\Pr(X>Y)-\Pr(X<Y)]^2. \end{align}$