I'm having trouble solving the following problem:
Suppose you have a unit disk (x^2 + y^2 <= 1), and you pick a point randomly from the disk, then let X be the x coordinate and y be the Y coordinate. Are X and Y independent?
I'm pretty sure the joint distribution is 1/pi when x^2 + y^2 <= 1
And I know in order to prove independence you have to multiply the marginal distributions, but my problem is I dont know what I have to integrate over to get the marginal distributions. Any help would be greatly appreciated.
No, X and Y are not independent. If Y = 0, then X can be in [0, 1], whereas if Y = 1, X = 0 necessarily.
A quick check on dependence in situations like this is to note that independent random variables always have support that is a Cartesian Product of sets i.e. the possible values of X cannot depend on Y and vice versa.