The random variables Y1 and Y2 are independent, both with density f (y) = 1 y2 , 1 < y
Chapter 6, Problem 6.69(choose chapter or problem)
The random variables Y1 and Y2 are independent, both with density f (y) = 1 y2 , 1 < y, 0, otherwise. Let U1 = Y1 Y1 + Y2 and U2 = Y1 + Y2. a What is the joint density of Y1 and Y2? b Show that the joint density of U1 and U2 is given by fU1,U2 (u1, u2) = 1 u2 1(1 u1)2u3 2 , 1/u1 < u2, 0 < u1 < 1/2 and 1/(1 u1) < u2, 1/2 u1 1, 0, otherwise. c Sketch the region where fU1,U2 (u1, u2) > 0. d Show that the marginal density of U1 is fU1 (u1) = 1 2(1 u1)2 , 0 u1 < 1/2, 1 2u2 1 , 1/2 u1 1, 0, otherwise. e Are U1 and U2 are independent? Why or why not?
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