Let (Y1, Y2) have joint density function fY1,Y2 (y1, y2) and let U1 = Y1 + Y2 and U2 =
Chapter 6, Problem 6.66(choose chapter or problem)
Let (Y1, Y2) have joint density function fY1,Y2 (y1, y2) and let U1 = Y1 + Y2 and U2 = Y2. a Show that the joint density of (U1, U2) is fU1, U2 (u1, u2) = fY1,Y2 (u1 u2, u2). b Show that the marginal density function for U1 is fU1 (u1) = " fY1,Y2 (u1 u2, u2) du2. c If Y1 and Y2 are independent, show that the marginal density function for U1 is fU1 (u1) = " fY1 (u1 u2) fY2 (u2) du2. That is, that the density of Y1 + Y2 is the convolution of the densities fY1 () and fY2 ()
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