STAT410 mid-term Practice1

# STAT410 mid-term Practice1 - 1 Let X and Y have the joint...

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1. Let X and Y have the joint probability density function f X , Y ( x , y ) = 10 x y 2 , 0 < x < y < 1, zero otherwise. a) Let a > 1. Find P ( Y > a X ). b) Find = > 2 1 Y 3 1 X P . c) Find = > 2 1 X 3 1 Y P . d) Find E ( X | Y = y ). e) Find E ( Y | X = x ). 2. Let X and Y have the joint probability density function f X , Y ( x , y ) = 10 x y 2 , 0 < x < y < 1, zero otherwise. Let U = Y – X and V = X + Y. Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ). 3. Let X and Y have the joint probability density function f X , Y ( x , y ) = 10 x y 2 , 0 < x < y < 1, zero otherwise. Let U = Y / X and V = X Y. Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ).

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4. Let X and Y have the joint probability density function f X , Y ( x , y ) = x 1 , x > 1, 0 < y < x 1 , zero elsewhere. a) Find f Y ( y ). b) Find f Y | X ( y | x ). c) Find E ( X ). d) Find E ( X | Y = y ). 5. Let X and Y have the joint probability density function f X , Y ( x , y ) = x 1 , x > 1, 0 < y < x 1 , zero elsewhere. Let U = Y and V = Y / X. Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ). 6. Let X and Y have the joint probability density function f X , Y ( x , y ) = x 1 , x > 1, 0 < y < x 1 , zero elsewhere. Let V = Y / X. Find the p.d.f. of V, f V ( v ).
7. Let X and Y have the joint probability density function f X , Y ( x , y ) = x 1 , x > 1, 0 < y < x 1 , zero elsewhere. Let U = X Y. Find the p.d.f. of U, f U ( u ). 8. Let X and Y have the joint probability density function f X , Y ( x , y ) = x 1 , x > 1, 0 < y < x 1 , zero elsewhere. Let W = X + Y. Find the p.d.f. of W, f W ( w ). 9. Let S and T have the joint probability density function f S , T ( s , t ) = t 1 , 0 < s < 1, s 2 < t < s . a) Find f S ( s ) and f T ( t ). b) Find E ( S ) and E ( T ). c) Find f S | T ( s | t ) and f T | S ( t | s ). d) Find E ( S | T = t ) and E ( T | S = s ). e) Find the correlation coefficient ρ S T . 10. Consider two continuous random variables X and Y with joint probability density function f ( x , y ) = 6 ( 1 – x ), 0 < y < x < 1. a) Find f X | Y ( x | y ). b) Find = > 2 1 Y 4 3 X P . c) Find = 2 1 Y X E . d) Are X and Y independent? e) Find Cov ( X, Y ).

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11. Consider two continuous random variables X and Y with joint probability density function f ( x , y ) = 6 ( 1 – x ), 0 < y < x < 1. Let U = X + Y and V = X – Y. Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ). 12.
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## This note was uploaded on 01/15/2012 for the course STAT 410 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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STAT410 mid-term Practice1 - 1 Let X and Y have the joint...

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