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# Hw08ans - STAT 408 Spring 2009 Homework#8(due Friday March...

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STAT 408 Spring 2009 Homework #8 (due Friday, March 20, by 3:00 p.m.) 1. Suppose that X follows a uniform distribution on the interval [ π / 2 , π / 2 ] . Find the c.d.f. and the p.d.f. of Y = tan X. f X ( x ) = < < - o.w. 0 2 2 1 π π π x F X ( x ) = < - + - < 2 1 2 2 2 1 2 0 π π π π π x x x x F Y ( y ) = P ( Y y ) = P ( tan X y ) = P ( X arctan ( y ) ) = ( 29 2 1 1 arctan + y π , < y < . f Y ( y ) = ( 2 1 1 y + π , < 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 π = ( 2 1 1 y + π , < y < . F Y ( y ) = ( + - y du u 2 1 1 π = ( 29 2 1 1 arctan + y π , < y < .

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2. An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder’s loss, Y , follows a distribution with density function: f ( y ) = otherwise 0 1 if 2 3 y y a) What is the expected value and the variance of the policyholder’s loss? E ( Loss ) = 1 3 2 dy y y = 1 2 - y = 2 . E ( Loss 2 ) = 1 3 2 2 dy y y = 1 2 ln - y is not finite. Var ( Loss ) is not finite. b) What is the expected value and the variance of the benefit paid under the insurance policy?
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