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Unformatted text preview: μ 1 = m 1 = ⇒ λ = 1 n n ± i =1 X i . Therefore, ˆ λ , our estimator of λ , is ˆ λ = 1 n n ± i =1 X i . Question 4. If X ∼ Binomial(7 , p ), then E { X } = 7 p . Set m 1 = 1 n ∑ n i =1 X i , μ 1 = 7 p . All we need to do is to set m 1 = μ 1 and solve for the unknown parameter. μ 1 = m 1 = ⇒ 7 p = 1 n n ± i =1 X i . 1 Therefore, ˆ p , our estimator of p , is ˆ p = 1 7 n n ∑ i =1 X i . Question 5. Let m 1 = 1 n n ∑ i =1 X i and m 2 = 1 n n ∑ i =1 X 2 i . Set m 1 = ˆ n ˆ p and m 2 = m 2 1 + ˆ n ˆ p (1ˆ p ). Solving this system of equations yields: m 1 = ˆ n ˆ p = ⇒ ˆ n = m 1 ˆ p m 2 = m 2 1 + ˆ n ˆ p (1ˆ p ) = m 2 1 + m 1 ˆ p ˆ p (1ˆ p ) = ⇒ ˆ p = 1m 2m 2 1 m 1 = m 1( m 2m 2 1 ) m 1 Therefore, ˆ n = m 1 ˆ p = m 2 1 m 1( m 2m 2 1 ) (Note that whenever the sample mean is smaller than the sample variance, ˆ n and ˆ p both become negative.) 2...
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This note was uploaded on 03/08/2012 for the course ORIE 4580 at Cornell.
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