411_hw04_soln

411_hw04_soln - Stat 411 Homework 04 Solutions 1. Problem...

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Unformatted text preview: Stat 411 Homework 04 Solutions 1. Problem 6.2.7 in the text . The PDF for the Gamma (4 , ) distribution is f ( x ) = 1 6 4 x 3 e- x/ , x > , > . (a) For the Fisher information, we first need second derivative of log-PDF: 2 2 log f ( x ) = 2 2 h const- 4 log - x i = 4 2- 2 x 3 . If we recall that the expected value of a Gamma ( , ) random variable is (see middle of p. 151 in the text), then I ( ) =- E h 2 2 log f ( X ) i = E (2 X ) 3- 4 2 = 2 4 2- 4 2 = 4 2 . (b) If X 1 ,...,X n iid Gamma (4 , ), then the MLE is found by maximizing the log- likelihood: ( ) = log L ( ) = const- 4 n log - n x/. Setting the derivative equal to zero and solving for gives: ( ) =- 4 n + n x 2 set = 0 = x 4 . If we recall that the variance of a Gamma ( , ) random variable is 2 (see middle of p. 151 in the text), then V ( ) = V ( X 1 ) 16 n = 4 2...
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This note was uploaded on 03/12/2012 for the course STAT 411 taught by Professor Staff during the Spring '08 term at Ill. Chicago.

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411_hw04_soln - Stat 411 Homework 04 Solutions 1. Problem...

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