IE4521_practice_midterm2

# IE4521_practice_midterm2 - 1 Problem 7.2.8 in the text...

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Unformatted text preview: 1. Problem 7.2.8 in the text. Solution (a) The bias of the estimate is bias = E (ˆ p )- p = E ( X/ 11)- p = 10 11 p- p =- p/ 11 (b) The variance of the estimate is Var(ˆ p ) = Var( X/ 11) = 1 121 Var( X ) = 10 p (1- p ) 121 (c) The mean square error is MSE = E (ˆ p- p ) 2 = Var(ˆ p ) + bias 2 = 10 p (1- p ) 121 + p 2 121 = 10 p- 9 p 2 121 (d) The bias of the estimator ˆ p 1 = X/ 10 is E (ˆ p 1 )- p = E ( X/ 10)- p = 0 The mean square error is therefore MSE(ˆ p 1 ) = Var(ˆ p 1 ) + bias 2 = 1 100 Var( X ) = p (1- p ) 10 The di erence between mean square errors is given by 10 p- 9 p 2 121 | {z } MSE for ˆ p- p (1- p ) 10 | {z } MSE for ˆ p 1 = 31 p 2- 21 p 1210 we want to know when the quantity 31 p 2- 21 p 1210 is negative, which only happens when 31 p 2- 21 p is negative; in other words, 31 p 2- 21 p ≤ p (31 p- 21) ≤ 31 p- 21 ≤ p ≤ 21 / 31 2. Suppose that a 6-sided die is rolled n times, showing values X 1 ,...,X n . Make a point estimate of the probability of rolling a 1 on the die, and write the standard error of this estimate. 1 Solution The number of 1 's shown on the die after n rolls is a B ( n,p = 1 / 6) random variable, which we can denote by Y . From lecture 11, slides 7-9, our estimate is the sample proportion ˆ p = Y/n , which has the approximate distribution ˆ p ∼ N p, p (1- p ) n = N 1 / 6 , 5 36 n 3. Determine the maximum likelihood estimator of θ when X 1 ,...,X n is a random variable with density function f ( x ) = 1 2 e-| x- θ | for-∞ < x < ∞ . Solution Write the likelihood function L ( x 1 ,...,x n ; θ ) = 1 2 e-| x 1- θ | ··· 1 2 e-| x n- θ | = 2- n e- ( | x 1- θ | + ··· + | x n- θ | ) The log-likelihood function is ‘ ( x 1 ,...,x n ; θ ) = log 2- n e- ( | x 1- θ | + ··· + | x n- θ | ) =- n log 2- | x 1- θ | - ··· - | x n- θ | We want to maximize this function, which is a bit sticky because of the absolute values. Note that d dθ | x i- θ | = (- 1 if θ < x i 1 if θ > x i and therefore d‘ dθ = ( #entries where θ > x i )- ( #entries where θ < x i ) To maximize this function we set...
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## This note was uploaded on 02/19/2011 for the course IE 4521 taught by Professor Staff during the Spring '08 term at Minnesota.

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IE4521_practice_midterm2 - 1 Problem 7.2.8 in the text...

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