I don't understand your distribution.
Why not just write P(X=0,Y=0)=... you only seem to have 3 points.
X
-1 0 1
-1 0 0 (1/3)
Y 0 0 (1/3) 0
1 (1/3) 0 0
Now for this i am pretty sure my calculations are correct but i want to check because there seems to be some errors in the textxbook which confused me.
for the marginal PMF say of x, i simply thought i simply sum the coresponding column.
For example p{-1)=1/3
p(0)=1/3
p(1)=1/3
But when they do this in the book the value they obtain does not corespond to the sum of the column they are refering to, and this happens twice, and i dont think there would be two errors in the same question and hence i'm a bit confused now.
Next part to find E[XY] i multiply all values of x and y which gives me XY=-1,0,1, with corresponding probabilities.
Then to check if these random variables are independendant i check if E[XY]=E[x]E[Y]
which in this case is zero. so they must be independant. Can someone tell me if i am doing something wrong.