Recitation8Solutions

Recitation8Solutions - n } x )) d x = -Z P (max { X 1 , . ....

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Steven Weber Dept. of ECE Drexel University ENGR 361: Statistical Analysis of Engineering Systems (Spring, 2008) Recitation 8 Solutions (Friday, May 23) Question 1. An rv X has CDF: F ( x ) = 0 , x < 0 ( x θ ) 2 , 0 x θ 1 , x > 1 . (1) A random sample ( X 1 , . . . , X n ) is gathered. The statistic ˆ θ = max { X 1 , . . . , X n } is proposed as a point estimator for the unknown parameter θ . Please do the following: Determine if the point estimator ˆ θ is biased ( E [ ˆ θ ] 6 = θ ) or unbiased ( E [ ˆ θ ] = θ ). If ˆ θ is biased, propose an unbiased point estimator. Prove your point estimator is unbiased. E [ ˆ θ ] = Z θ 0 P (max { X 1 , . . . , X n } > x ) d x = Z θ 0 (1 - P (max { X 1 , . . . , X
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Unformatted text preview: n } x )) d x = -Z P (max { X 1 , . . . , X n } x ) d x = -Z P ( X 1 x ) P ( X n x )d x = -Z F ( x ) n d x = -Z x 2 n d x = -1 2 n Z x 2 n d x = -x 2 n +1 (2 n + 1) 2 n = - 2 n + 1 = 2 n 2 n + 1 . (2) The estimator is biased. We propose: = 2 n +1 2 n . It is unbiased: E [ ] = 2 n + 1 2 n E [ ] = 2 n + 1 2 n 2 n 2 n + 1 = . (3) www.ece.drexel.edu/faculty/sweber 1 May 23, 2008...
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