hw6 - MATH 471 HW 6 Solutions Pengsheng Ji TA Oce Hours...

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MATH 471 HW 6 Solutions Pengsheng Ji TA Office Hours: 4:05-5:05 PM Tuesday 5:20-6:20 PM Thursday 218 Mallot Hall 3.3.5 X has density function f ( x ) = λe - λx and r ( x ) = x 1 has the inverse s ( y ) = y α . By equation (3.1) the density function of r ( X ) is given by f ( s ( y )) · s 0 ( y ) = λe - λy α · αy α - 1 , for y 0 . 3.3.11 (a)For y > 0 , P ( Y y ) = P ( - y X y ) = F ( y ) - F ( - y ) . Differentiating it on y , we get the density function of Y is f ( y ) + f ( - y ) for 0 < y < 1 and 0 elsewhere. (b) For y > 0 , P ( Y y ) = P ( - y X y ) = F ( y ) - F ( - y ) . By differentiating it on y , we get the density of Y is [ f ( y ) + f ( - y )] · 1 2 · y - 1 / 2 , for 0 < y < 1 and 0 elsewhere. 3.4.10 P ( X + Y < 1) = Z 1 0 Z 1 - x 0 6 xy 2 dydx = Z 1 0 2 x (1 - x ) 3 dx = 1 / 20 3.4.12 The event that no piece is longer than 1/2 is

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