hw9soln - ORIE 3500/5500, Fall 10 HW 9 Solutions Assignment...

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Unformatted text preview: ORIE 3500/5500, Fall 10 HW 9 Solutions Assignment 9 Solutions Problem 1 Clearly, Y takes values in [0,1]. So F Y ( y ) = 0 for y 0 and F Y ( y ) = 1 for y 1. Now for y (0 , 1) F Y ( y ) = P ( Y y ) = P ( X y or 1 X y ) = P ( X y ) + P ( X 1 y ) = 1- e- y + (1- (1- e- 1 y )) = 1- e- y + e- 1 y Problem 2 Define a function T : R \{ } 7 R \{ } as T ( x ) = x, if x > ,- x 2 / 4 , if x < . So, T is a one-one function with inverse function given by T- 1 ( y ) = y, if y > ,- - 4 y, if y < . Also, note that X is a continuous random variable and hence, P ( X = 0) = 0. So, the density of Y = T ( X ) is given by f Y ( y ) = f X ( T- 1 ( y )) dT- 1 ( y ) dy = (1- y )1 { <y< 1 } 1 , if y > , (1- - 4 y )1 {- 1 <- - 4 y< } 1 - y , if y < . = (1- y ) , if 0 < y < 1 , ((- y )- 1 / 2- 2) , if- 1 4 < y < . Problem 3 The table below presents values of Z for certain values of X and Y X/Y 1 2 3 1 2 3 1 1 2 3 4 2 2 3 4 5 3 3 4 5 6 4 4 5 6 7 5 5 6 7 8 1 ORIE 3500/5500, Fall 10 HW 9 Solutions So P ( Z = 0) = P ( X = 0 ,Y = 0) = . 4 P ( Z = 1) = P ( X = 1 ,Y = 0) + P ( X = 0 ,Y = 1) = . 38 Continue adding up, we get z 1 2 3 4 5 6 7 8 P ( Z = z ) .4 .38 .1 .054 .029 .021 .011 .004 .001 Problem 4 (a) Let us define a one-one transformation Tr : ( x,y ) 7 ( xy,x/y ) defined on (0 , ) (0 , ) (Note the space where the function Tr is defined). Then, ( Tr...
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This note was uploaded on 10/29/2011 for the course MATH 3310 at Cornell University (Engineering School).

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hw9soln - ORIE 3500/5500, Fall 10 HW 9 Solutions Assignment...

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