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4. A point is chosen at random (according to a uniform PDF) within a semicircle of...

4. A point is chosen at random (according to a uniform PDF) within a semicircle of the form
{(x, y)|x2 + y2 <= r2, y >= 0}, for some given r > 0.

(a) Find the joint PDF of the coordinates X and Y of the chosen point. [6 points]
(b) Find the marginal PDF of Y and use it to find E[Y ]. [7 points]
(c) Check your answer in (b) by computing E[Y ] directly without using the marginal PDF of
Y . [7 points]

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