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Unformatted text preview: STAT 409 Homework 1 ( Answers ) Fall 2008 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( ) ( ) ( ) & 2 X X ln 1 & & ; x x x f x f ⋅ = = , x > 1, θ > 1. a) Find the maximum likelihood estimator & ˆ of θ . L( θ ) = ( ) ∏ = ⋅ n i i i x x 1 & 2 ln 1 & . ln L( θ ) = ( ) & & = = ⋅ + n i i n i i x x n 1 1 ln & ln ln 1 & 2 ln . ( ) & = = n i i d d x n 1 ln 1 & 2 & & L = 0. ¡ & = + = n i i x n 1 ln 2 1 & ˆ . b) Suppose θ > 2. Find the method of moments estimator & ~ of θ . E(X) = ( ) ( ) ( ) ( ) 2 2 1 & 2 X 2 & 1 & ln 1 & = = ¢ ¢ ∞ ∞ ∞ ⋅ ⋅ ⋅ dx x x x dx x f x . ( ) ( ) 2 2 1 2 & 1 & 1 = = & = ⋅ x x n n i i . ¡ 1 1 2 & ~ = x x . 2. Fortyeight measurements are recorded to several decimal places. Each of these 48 numbers is rounded off to the nearest integer. The sum of the original 48 numbers is approximated by the sum of these integers. If we assume that the errors made by rounding off are i.i.d. and have uniform assume that the errors made by rounding off are i....
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This note was uploaded on 10/11/2010 for the course STATS 410 taught by Professor Alexey during the Spring '10 term at University of Illinois at Urbana–Champaign.
 Spring '10
 Alexey
 Probability

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