# 410Hw08ans - STAT 410 Homework #8 (due Wednesday, July 22,...

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STAT 410 Homework #8 Summer 2009 (due Wednesday, July 22, by 4:00 p.m.) Warm-up: 4.3.7 Hint: ( 29 ( 29 α β 1 1 M Gamma t t - = , t < 1 / . M Y n ( t ) = E ( e t X n / n ) = M X n ( t / n ) = n n t 1 1 - e t as n . Let X be a “random” variable with P ( X = ) = 1. ( degenerate distribution at ) Then M X ( t ) = E ( e t X ) = e t . X Y D n . 1. 4.3.2 F Y 1 ( x ) = ( 29 θ - - - x n e 1 , x > . ( ) F Z n ( z ) = P ( n ( Y 1 ) z ) = P ( Y 1 n z + ) = 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. )

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2. 4.3.4 Hint: F Y 2 ( x ) = ( 29 ( 29 ( 29 ( 29 = - - n i i n i x x i n 2 F 1 F = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 F 1 F F 1 1 - - - - - n n x x n x . F Y 2 ( x ) = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 F 1 F F 1 1 - - - - - n n x x n x . Since W n = n F ( Y 2 ), P ( 0 < W n < n ) = 1. Let 0 < w < n . F W n ( w ) = P ( n F ( Y 2 ) w ) = - n w 1 Y F F 2 = 1 1 1 1 - - - - - n n n w n w n n w 1 – e w w e w as n . For the limiting distribution W of W n , F W ( w ) = 1 – e w w e w , w > 0, f W ( w ) = w e w , w > 0. Therefore, W has Gamma distribution with α = 2, β = 1.
3. a) 4.3.9 Hint: We already know that ( 29 ( 29 1 , 0 2 Y N D n n n - . χ 2 ( v = 50 ) can be approximated by N ( v , 2 v ) = N ( 50 , 100 ). P ( 40 < X < 60 ) = P ( 10 50 40 - < Z < 10 50 60 - ) = P ( – 1.00 < Z < 1.00 ) = 0.8413 – 0.1587 = 0.6826 . b) Find P ( 40 < X < 60 ), where X has a ( 29 50 2 χ distribution. Hint: Use integration by parts 24 times or EXCEL: = CHIINV ( α , v ) gives ( 29 v 2 α χ for 2 χ distribution with v degrees of freedom, x s.t. P ( ( 29 v 2 χ > x ) = α . = CHIDIST ( x , v ) gives the upper tail probability for 2 χ distribution with v degrees of freedom, P ( ( 29 v 2 χ > x ) . P

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## 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.

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410Hw08ans - STAT 410 Homework #8 (due Wednesday, July 22,...

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