Let $(X,Y)$, a 2D stochastic variable with density function $f(x,y) = 3/2 xy1_A(x,y)$, where $A$ is the set of all positive values bound by the line $y = 2-x$.
Find the distribution of $(X + Y)$ and $(X - Y)$.
Things I know and thoughts about a solution.
I know that the variables are not independent as their J.D.F doesn't equal the product of the marginals distributions. I also know that a distribution function is equal to the integral of the density function but I'm not sure how this applies here. Am I supposed to sum the marginal distribution functions and integrate that to find the distribution?
I also know there is the possibility of transforming the distribution but again I'm not sure how to apply this. Do I take the reciprocal of the integral of the density function, if so with respect to which variable? There is also a consideration of limits too I suppose.