Hw02ans - STAT 410 Fall 2008 Homework #2 (due Friday,...

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Unformatted text preview: STAT 410 Fall 2008 Homework #2 (due Friday, September 12, by 3:00 p.m.) 1. ( ~ 1.9.19 ) Let X be a nonnegative continuous random variable with p.d.f. f ( x ) and c.d.f. F ( x ). Show that E ( X ) = ( ) ( ) & -1 x x d F . E ( X ) = ( ) & x x f x d = ( ) & & x x f y d x d = ( ) & & x y x f d x d ( ) & & x y x f d x d = ( ) & & y x x f d y d E ( X ) = ( ) & & y x x f d y d = ( ) & > X P y y d = ( ) ( ) & -1 y y d F . OR ( ) ( ) & -1 x x d F = by parts: u = 1 F ( x ) dv = dx du = f ( x ) dx v = x = ( ) ( ) ( ) ( ) & ---1 x x f x x x d F = ( ) & x x f x d = E ( X ). 2. Suppose that X follows a uniform distribution on the interval [ / 2 , / 2 ] . Find the c.d.f. and the p.d.f. of Y = tan X. f X ( x ) = & & & & < <-o.w. 2 2 1 x F X ( x ) = & & & & & & < -+-< 2 1 2 2 2 1 2 x x x x F Y ( y ) = P ( Y y ) = P ( tan X y ) = P ( X arctan ( y ) ) = ( ) 2 1 1 arctan + y , < y < . f Y ( y ) = ( ) 2 1 1 y + , < y < . ( Standard ) Cauchy distribution. OR g ( x ) = tan x g 1 ( y ) = arctan ( y ) d x / d y = 2 1 1 y + f Y ( y ) = f X ( g 1 ( y ) ) y x d d = + 2 1 1 1 y = ( ) 2 1 1 y + , < y < . F Y ( y ) = ( ) +-y du u 2 1 1 = ( ) 2 1 1 arctan + y , < y < . 3. An insurance policy reimburses a loss up to a benefit limit of 10. The policyholders loss, Y, follows a distribution with density function: f ( y ) = & & & & > otherwise 1 if 2 3 y y What is the expected value and the variance of the benefit paid under the insurance policy? The benefit paid under the insurance policy = & & < 10 for 10 10 1 for y y y E ( Benefit Paid ) = + 10 3 10 1 3 2 10 2 dy y dy y y = 10 2 1 10 10 2 --y y = 1.9. E ( Benefit Paid 2 ) = + 10 3 2 10 1 3 2 2 10 2 dy y dy y y = 10 2 1 10 100 2 ln --y y = 2 ln 10 + 1. Var ( Benefit Paid ) = 2 ln 10 + 1 1.9 2 = 2 ln 10 2.61 1.99517. 4. The time, T, that a manufacturing system is out of operation has cumulative distribution function F ( t ) = & & & & > -otherwise 2 if 2 1...
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Hw02ans - STAT 410 Fall 2008 Homework #2 (due Friday,...

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