New types of random variables

[fierce thistle]

I missed a few classes and now I cannot handle this notation of superscripting with the plus and minus signs. I think it's purpose is to ensure non-negativity of random variables. Please guide. Also how is this related to the general definition of expectation.

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[crimson flume]

Im not familiar with this notation but if I am guessing if X is a real random variable on a probability space (G,T,P) if you define for w in G
X+(w)=max(0,X(w)) and
X-(w)=min(0,X(w)) then you have X=X+ + X- but I’m not sure

[elder bay]

If the purpose is to ensure non-negativity then it is likely that X^-(w) = max(0, -X(w))
Similar treatment is used in linear programming

and X = X^+ - X^-

[crimson flume]

Yes I was thinking so as well but it seemed kinda odd considering the properties here

[elder bay]

Are there some properties of (X, Y) that you have not told us so that the first four properties hold, e.g. they are independent?