# Hw11ans - STAT 408 Spring 2009 Homework#11(due Thursday...

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STAT 408 Spring 2009 Homework #11 (due Thursday, April 16, by 3:00 p.m.) 4.4-8

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4.4.14 F Z ( z ) = P ( Z z ) = P ( X z Y ) = - - 0 5 5 2 5 1 5 dx dy x z x y x e e = - - 0 5 5 2 5 dx x z x x e e = ( 29 + - 0 5 1 2 5 dx x z z x e = ( 29 2 2 1 + z z , z > 0. f Z ( z ) = F Z ' ( z ) = ( 29 3 1 2 + z z , z > 0. OR
1. Let X and Y have the joint probability density function f X, Y ( x , y ) = < < < + otherwise 0 1 0 4 x y y x Let U = X Y and V = X. Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ). X = V, Y = V U . 0 < y 0 < u , y < x u < v 2 , x < 1 v < 1. J = 1 1 0 2 v u v - = v 1 - . | J | = v 1 . f U, V ( u , v ) = f X, Y ( v , v u ) | J | = v v u v 1 4 + = 2 4 1 v u + , 0 < v < 1, 0 < u < v 2 , f U, V ( u , v ) = 0 otherwise.

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2. Let X and Y have the joint p.d.f. f X Y ( x , y ) = 20 x 2 y 3 , 0 < x < 1, 0 < y < x . Let U = Y 2 and V = X Y. Find the joint probability density function of ( U, V ), f U V ( u , v ). Sketch the support of ( U, V ). Y = U X = U V 0 < y y < x u 3 / 2 < v , x < 1 v < u . J
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## This note was uploaded on 04/27/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Hw11ans - STAT 408 Spring 2009 Homework#11(due Thursday...

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