Let A > 0. Suppose (X, Y) has joint probability density function given, for all x, y E R, by 3. fx.x (2,y) =0, { Ay exp(-y(x + )), if x > 0 and y > 0 otherwise. (i) Calculate the marginal density of Y. (ii) Let y > 0. Find the the conditional probability density function of X given Y = y. Interpret your answer. (iii) Calculate E[X|Y] and prove that E[X] = +oo. Hint: You can use without a proof that dy = +o.

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A: It is an important part of statistics. It is widely used.