Two random variables X,Y are, X,Y independant b. are X+Y X-Y independant

Question

if X and Y are independent, check whether the (0, 0) value is the same as P(X=0) P(Y= 0), and the same with the other 4 entries.

Make a table with the distributions of X + Y and X - Y. For any values of x and y, that creates an entry in the X + Y, X- Y table. For instance, X = 0, Y = 0 would give you X + Y = 0, X - Y = 0. So enter a 0.5 in that position. X - 0, Y = 1 gives X + Y = 1, X - Y = -1. Put a 0.2 in that position.

Im stuck on the second part rv X+Y is 0,1,2 rv X-Y is -1,0,1 but Im not able to fill values for (2,1)(2,-1)(0,-1)(0,1)(1,0) please help this is a hw problem

Answer 1

If $X+Y = 2$ and $X-Y = 1$, then we must have $X = 1.5$ and $Y = 0.5$.

Is that even a possible outcome? If not, then $\Pr[X+Y = 2 \ \text{AND} X-Y = 1] = 0$.

Note: In general, if $X$ and $Y$ are integers, then $X+Y$ and $X-Y$ are either both even or both odd because their difference $(X+Y)-(X-Y) = 2Y$ must be even.