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4. Let X and Y be random variables having joint probability density function (pdf) f(x, y)...

4. Let X and Y be random variables having joint probability density function (pdf) f(x, y) = 4/7 (xy − y), 4 < x < 5 and 0 < y < 1

(a) Find the marginal density fY (y).

(b) Show that the marginal density, fY (y), integrates to 1 (i.e., it is a density.)

(c) Find fX|Y (x|y), the conditional density of X given Y = y.

(d) Show that fX|Y (x|y) is actually a pdf (i.e., it integrates to 1.) Find IE(Y ).

(e) State, but do not evaluate, the integral whose value is IE(e^XY ).

(f) Are X and Y independent? Explain. (You do not need to do any additional calculations to answer this question.)

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