8. (10 pts) Let (X, Y) be a point selected at random from the upper half-disk with radius 1 according to a uniform distribution. In other words, the joint PDF of X and Y is given by f_{X,Y}(x,y) = {2/?, for x^2 + y^2 < 1 and y > 0, 0, elsewhere. Also, define the random variables R and T by R = sqrt{X^2 + Y^2} and T = X / Y. (a) Find the PDF of R. (b) Find the joint PDF of R and T. (c) Are R and T independent?
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The CDF of R is given by F_R(r) = P(R <= r) = P(X^2 + Y^2 <= r^2) for r in [0,1]. Show more…
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