410Hw03ansJeff - STAT 410 Fall 2011 Homework#3(due Friday...

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
STAT 410 Fall 2011 Homework #3 (due Friday, September 16, by 3:00 p.m.) 1. 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 , zero elsewhere. a) Find f X | Y ( x | y ) . Recall: f Y ( y ) = ( ) 9 3 3 20 y y - , 0 < y < 1. f X | Y ( x | y ) = ( ) ( ) y f y x f , Y Y X, = 6 2 1 3 y x - , y 2 < x < 1. b) Find E ( X | Y = y ) . E ( X | Y = y ) = - 1 6 2 2 1 3 y dx y x x = 6 8 1 1 4 3 y y - - , 0 < y < 1. c) Find f Y | X ( y | x ) . Recall: f X ( x ) = 5 x 4 , 0 < x < 1. f Y | X ( y | x ) = ( ) ( ) x f y x f , X Y X, = 2 3 4 x y , 0 < y < x . d) Find E ( Y | X = x ) . E ( Y | X = x ) = x dy x y y 0 2 3 4 = x 5 4 , 0 < x < 1.
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2. Once a car accident is reported to an insurance company, the company makes an initial estimate, X, of the amount it will pay to the claimant. When the claim is finally settled, the company pays an amount, Y, to the claimant. The company has determined that X and Y have the joint p.d.f. f ( x , y ) = ( ) ( ) ( ) 1 1 2 2 1 2 - - - - x x y x x , x > 1, y > 1. a) Given that the initial claim estimated by the company is 1.5, determine the probability that the final settlement amount exceeds 2. Recall: - 1 dy y b a = 1 - a b , a > 1. f X ( x ) = ( ) ( ) ( ) - - - - 1 1 1 2 2 1 2 dy y x x x x = ( ) 1 1 1 2 1 2 2 - - - - x x x x = 3 2 x , x > 1. f Y | X ( y | x ) = ( ) ( ) x f y x f , X = ( ) ( ) 1 1 2 1 - - - - x x y x x , y > 1. f Y | X ( y | x = 1.5 ) = 4 3 - y , y > 1. P ( Y > 2 | X = 1.5 ) = - 2 4 3 dy y = 8 1 = 0.125 . b) Find E ( Y | X = x ) . E ( Y | X = x ) = ( ) ( ) - - - - 1 1 1 2 1 dy y x x y x x = ( ) - - - 1 1 1 dy y x x x x = 1 1 1 - - - x x x x = x , x > 1.
Image of page 2
3. When you leave your car at Honest Harry’s Car Repair Shop , first it takes X weeks for needed parts to arrive, and then Y more weeks for the repairs to be finished. Thus the total wait is W = X + Y weeks. Suppose that X and Y are independent, the p.d.f. of X is f X ( x ) = 2 x , 0 < x < 1, zero otherwise, and Y has a Uniform distribution on interval ( 0, 1 ) . Find the p.d.f. of W, f W ( w ) = f X + Y ( w ) . 0 < x < 1, 0 < y < 1 0 < x + y < 2 ( ) ( ) ( ) - = - + dx x w f x f w f Y X Y X . ( ) < < = otherwise 0 1 0 2 X x x x f ( ) < < = otherwise 0 1 0 1 Y y y f ( ) < < - = < - < = - otherwise 0 1 1 otherwise 0 1 0 1 Y w x w x w x w f Case 1. 0 < w < 1 w – 1 < 0 Case 2. 1 < w < 2 0 < w – 1 < 1 ( ) ( ) = + w dx x w f 0 Y X 1 2 = w 2 , 0 < w < 1. ( ) ( ) - + = 1 1 Y X 1 2 w dx x w f = 1 – ( w – 1 ) 2 = 2 w w 2 , 1 < w < 2. OR
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
( ) ( ) ( ) - = - + dy y f y w f w f Y X Y X . Case 1. 0 < w < 1 w – 1 < 0 Case 2. 1 < w < 2 0 < w – 1 < 1 ( ) ( ) - = + w dy y w w f 0 Y X 2 1 = w 2 , 0 < w < 1.
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern