let U and V independent variables that are both defined on [-∏, ∏] and are uniformly distributed.
If x = cos(U + V) and y = sin(U-V), what is the area where the variables X and Y are defined?
U + V = arccos(x)
U - V = arcsin(y)
For a test if we have chosen the right area, we can deploy probability density function f_(x,y) = f(u(x,y),v(x,y)*J(x,y), where J(x,y) is a Jacobian determinant.
The Attempt at a Solution
My attempt: x and y have minimum and maximum at -1 and 1, respectively.
I would be really grateful if anyone could help me, since this exercise is beyond my level of knowledge. If I have to remove the exercise to any other forum, feel free to remove the issue. Wish you all a pleasant day!:)