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Unformatted text preview: 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. 2 2 1 π π π x F X ( x ) = ≥ < ≤ + < 2 1 2 2 2 1 2 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 ) = ( 29 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 = ( 29 2 1 1 y + π , – ∞ < y < ∞ . F Y ( y ) = ( 29 ∫ ∞ + y du u 2 1 1 π = ( 29 2 1 1 arctan + y π , – ∞ < y < ∞ . 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 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....
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This note was uploaded on 04/27/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 STEPANOV
 Statistics

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