Proabibility - Random variables independence question

serhannn

1. The problem statement, all variables and given/known data
Two variables, X and Y have a joint density f(x,y) which is constant (1/∏) in the circular region x²+y² <= 1 and is zero outside that region
The question is: Are X and Y independent?

2. Relevant equations
Well, I know that for two random variable to be independent, multiplication of their marginal densities must equal their joint denstiy, i.e.
f(x,y)=f_X(x)*f_Y(y)

3. The attempt at a solution
My problem is I am confused about how to select the integration limits. I know how to do it when simple boundaries are given (like x<2 and y>1, etc.) but within a circular region, I just could not figuer out how to do it. Should I, for example, integrate y from -sqrt(1-x² ) to sqrt(1-x²) or is that a wrong approach? How can I select the integration limits in a circular region? Is it a better approach to convert to polar coordinates first and then integrate?

Thanks a lot for your help.

clamtrox

To be independent means that if I give you value of x, that doesn't increase your knowledge on value of y. If I tell you that x=1, can you tell me anything about y?