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Hw09ans - STAT 410(due Monday March 31 by 4:30 p.m...

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STAT 410 Homework #9 Spring 2008 (due Monday, March 31, by 4:30 p.m.) No credit will be given without supporting work. Warm-up: 4.2.3 By Chebyshev’s Inequality, P ( | W n μ | ε ) 2 2 W ° ± n = 2 ° p n b 0 as n for all ε > 0. Therefore, ² W P n . 1. 4.3.2 F Y 1 ( x ) = ( ) θ - - - x n e 1 , x > θ . ( Recall Homework 8, problems 6 & 7. ) F Z n ( z ) = P ( n ( Y 1 θ ) z ) = P ( Y 1 n z + θ ) = 1 – e z , z > 0. Therefore, the limiting distribution of Z n is Exponential with mean 1. ( Exponential distribution with mean 1 is same as Gamma distribution with α = 1, β = 1. ) 2. 4.3.3 F Y n ( x ) = ( ) ( ) n x F . Since Z n = n ( 1 – F ( Y n ) ) , P ( Z n 0 ) = 1. Let z > 0.
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F Z n ( z ) = P ( n ( 1 – F ( Y n ) ) z ) = P ( F ( Y n ) 1 – n z ) = ° ° ± ² ³ ³ ´ µ ° ± ² ³ ´ µ - - - n z n 1 F F 1 1 Y = n n z 1 1 ° ± ² ³ ´ µ - - 1 – e z as n . For the limiting distribution Z of Z n , F Z ( z ) = 1 – e z , z > 0, f Z ( z ) = e z , z > 0. Therefore, the limiting distribution of Z n is Exponential with mean 1. ( Exponential distribution with mean 1 is same as Gamma distribution with α = 1, β = 1.
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