Let \(B\) be the upper half of the circle \(x^{2}+y^{2}=1\). The random vector \((X, Y)\) is uniformly distributed over \(B\).

(1) Determine the joint density of \((X, Y)\).

(2) Determine the marginal distribution densities.

(3) Are \(X\) and \(Y\) independent? Is theorem 3.1 applicable to answer this question?

Data from Theorem 3.1

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Theorem 3.1 (1) If X and Y are independent and in the respective intervals [a, b] and [c, d] uniformly distributed, then the random vector (X, Y) has a uniform distribution on the rectangle B={axb, c y d}. (2) Conversely, if (X,Y) has a uniform distribution on the rectangle B, then the ran- dom variables X and Y are independent and uniformly distributed in the intervals [a, b] and [c, d], respectively.