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Let U and V be two independent standard normal random variables, and let X = |U|...

Let U and V be two independent standard normal random variables, and let X = |U| and Y = |V|.

Let R = Y/X and D = Y-X.

(1) Find the joint density of (X,R) and that of (X,D).

(2) Find the conditional density of X given R and of X given D.

(3) Find the expectation of X given R and of X given D.

(4) Find, in particular, the expectation of X given R = 1 and of X given D=0.

(5) Why the two answers of (4) are different.

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