I saw this post correlation between positive and negative part of a Random Variable and NCh's answer.
Here $X$ is a standard normal RV, $X^{+}, X^{-}$ are its positive and negative parts.
I find it interesting that $X^- = -\min(X, 0)$ is more commonly used, which would mean that $X^- \geq 0$ and $E[X^-] \geq 0$. What is the reasoning behind this instead of defining $X^ – = \min(X, 0) \leq 0$?
Depending on which definition you use your correlation/covariance will have opposite signs.
In addition, if we use the definition that $X^- = -\min(X, 0) \geq 0$. Then how can $E[X^+ X^-] = 0$? If we use the definition $X^- = \min(X, 0) \leq 0$, then we can make a symmetry argument that its expectation of the product is zero.
Best Answer
It does not matter which definition you use so long as the reader is clear. Similarly with the complex number $2+3i$, you could call the imaginary part $3i$ or $3$ but you need to be clear which you are using.
It may be more convenient in this case to have $X^-$ having the same distribution as $X^+$, which would justify using the non-negative version and $X=X^+ - X^-$
In either definition $X^+ \not=0 \implies X^- =0$ and $X^- \not=0 \implies X^+ =0$ and the product is always $0$
So $\mathbb P(X^+X^-=0)=1$ and $\mathbb E[X^+X^-]=0$