Let [f(x, y)=frac{1}{2} sin (x+y), 0 leq x, y leq frac{pi}{2}] be the joint probability density of

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Let

\[f(x, y)=\frac{1}{2} \sin (x+y), 0 \leq x, y \leq \frac{\pi}{2}\]

be the joint probability density of the random vector \((X, Y)\).

(1) Determine the marginal densities.

(2) Are \(X\) and \(Y\) independent?

(3) Determine the conditional mean value \(E(Y \mid X=x)\).

(4) Compare the numerical values \(E(Y \mid X=0)\) and \(E(Y \mid X=\pi / 2)\) to \(E(Y)\). Are the results in line with your anwer to (2)?

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