# 410Hw04ans - STAT 410 Summer 2009 Homework#4(due Monday...

This preview shows pages 1–4. Sign up to view the full content.

STAT 410 Summer 2009 Homework #4 (due Monday, July 6, by 4:00 p.m.) 1. 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 ). f Y ( y ) = ( 29 - 1 1 6 y dx x = 3 – 6 y + 3 y 2 = 3 ( 1 – y ) 2 , 0 < y < 1. f X | Y ( x | y ) = ( 29 ( 29 2 1 1 2 y x - - , y < x < 1, 0 < y < 1. b) Find = 2 1 Y 4 3 X P . f X | Y ( x | 2 1 ) = 8 ( 1 – x ), 2 1 < x < 1. = 2 1 Y 4 3 X P = ( 29 - 1 4 3 1 8 dx x = 4 1 . c) Find = 2 1 Y X E . f X | Y ( x | 2 1 ) = 8 ( 1 – x ), 2 1 < x < 1. = 2 1 Y X E = ( 29 - 1 2 1 1 8 dx x x = 3 2 .

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

View Full Document
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 ) = ( 29 ( 29 ( 29 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. f X ( x ) = ( 29 ( 29 ( 29 - - - - 1 1 1 2 2 1 2 dy y x x x x = 3 2 x , x > 1. f Y | X ( y | x ) = ( 29 ( 29 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 ) = ( 29 ( 29 - - - - 1 1 1 2 1 dy y x x y x x = x , x > 1.
3. Let X denote the number of times a certain numerical control machine will malfunction: 0, 1, or 2 times, on any given day. Let Y denote the number of times a technician is called on an emergency call. The joint p.m.f. p X, Y ( x , y ) is presented in the table below: X Y 0 1 2 0 0.10 0.05 0.05 1 0.10 0.15 0.15 2 0 0.15 0.25 a) Find P ( Y > X ). P ( Y > X ) = p ( 1, 0 ) + p ( 2, 0 ) + p ( 2, 1 ) = 0.25 . b) Find p X ( x ), the marginal p.m.f. for the number of machine malfunctions. p X ( x ) = = = = 2 1 0 , 45 . 0 , 35 . 0 , 20 . 0 x x x c) Find p Y | X ( y | 2 ), the conditional p.m.f. for the number of emergency calls given two machine malfunctions. p Y | X ( y | 2 ) = ( 29 ( 29 2 , 2 X p y p = ( 29 45 . 0 , 2 y p = = = = 2 1 0 , 45 . 0 25

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 10/29/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

### Page1 / 14

410Hw04ans - STAT 410 Summer 2009 Homework#4(due Monday...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online