sum and product of random variables 
Let $X$ and $Y$ be two (general) random variables with finite means and let $Z=X+Y$ and $Z' = XY.$ Define for a random variable $X,X^+$ to be the random variable equal to $\max\{X(\omega), 0\}$ and $X^-$ so that $X^-(\omega) = \max\{-X(\omega), 0\},$ where $\omega \in \Omega,$ the sample space.


Express $Z^+$ and $Z^-$ in terms of $X^+, X^-, Y^+, Y^-,$ with justification.


If $X$ and $Y$ are independent, express $Z'^+$ and $Z'^-$ in terms of $X^+, X^-, Y^+, Y^-,$ with justification.

I think $E(Z^+) - E(Z^-) = E(X^+) - E(X^-) + E(Y^+) - E(Y^-).$ But I'm not sure how to express $Z^+$ and $Z^-$ from this. I tried considering all the possible linear combinations of $X^+, X^-, Y^+, Y^-,$ with coefficients of absolute value less than or equal to $1$, but that seems very tedious. For instance, clearly $Z^+\neq X^+ + Y^+ - X^- - Y^-$ because they're unequal when $X(\omega), Y(\omega) < 0.$ Is there some sort of way for me to deterministically find the right coefficients?
Also, if $X$ and $Y$ are independent, then $E(Z'^+) - E(Z'^-) = E(Z')= E(XY) = E(X)E(Y) = (E(X^+) - E(X^-))(E(Y^+)- E(Y^-))$, and so the values of $Z'^+$ and $Z'^-$ should reflect this.

Could someone give some hints as to how to find the required relationships?


Using all the possibilities that I believe are possible, e.g. when X, Y are both positive, when both negative, when X is positive and Y is negative but greater in absolute value or negative but less in absolute value, and with X and Y's roles switched for this last statement
X    Y    Z     X+   Y+   X-    Y-    Z+    Z-
1    2    3     1    2    0     0     3     0              
1    -1/2 1/2   1    0    0     1/2   1/2   0                  
-1/2 1    1/2   0    1    1/2   0     1/2   0                  
1    -2   -1    1    0    0     2     0     1             
-2   1    -1    0    1    2     0     0     1              
-1   -3   -4    0    0    1     3     0     4            

I found that
$$Z^+=\max\{X^+-Y^-, 0\}+\max\{Y^+-X^-, 0\}\\
Z^-=\max\{X^--Y^+, 0\}+\max\{Y^--X^+, 0\}$$
and e.g. both positive, exactly one of them negative, and both negative,
X    Y    Z'    X+   Y+   X-    Y-    Z'+    Z'-
1    2    2     1    2    0     0     2     0              
1    -1/2 -1/2  1    0    0     1/2   0     1/2                                 
-1    2   -2    0    2    1     0     0     2                      
-1   -3   3     0    0    1     3     3     0            

gives
$$Z'^+=X^+Y^++X^-Y^-\\
Z'^-=X^+Y^-+X^-Y^+$$
