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Suppose the continuous random variables X and Y have joint pdf: fXY (x, y) = （1/2）xy...

Suppose the continuous random variables X and Y have joint pdf: fXY (x, y) = （1/2）xy for 0 < x < 2 and x < y < 2 (a) Find P(X < 1, Y < 1). (b) Use the joint pdf to find P(Y > 1). Be careful setting up your limits of integration. (c) Find the marginal pdf of Y , fY (y). Be sure to state the support. (d) Use the marginal pdf of Y to find P(Y > 1). (e) Find E(Y ). (f) Find P(Y < 2X) by integrating in the x direction first. Be careful setting up your limits of integration. (g) Find P(Y < 2X) by integrating in the y direction first. Be extra careful setting up your limits of integration. (h) Find the conditional pdf of X given Y = y, fX|Y =y(x). Be sure to state the support (and the values y that can be conditioned on). (i) Find the expected value of X given Y = y, E(X|Y = y). (Be sure to state the values y that can be conditioned on.) (j) Using your answer from (1h), find the conditional pdf of X given Y = 1, fX|Y =1(x). Be sure to state the support. (k) Using your answer from (1i), find the expected value of X given Y = 1, E(X|Y = 1). (l) Find P(X < Y < √ 2X) by integrating in the x direction first. Be careful setting up your limits of integration. (m) Find E(X/Y ). (n) Find the marginal pdf of X, fX(x). Be sure to state the support. (o) Are X and Y independent? Why?

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