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Unformatted text preview: M4056 Quiz 6, Oct. 1, 2010 Name Recall that the Poisson distribution with parameter λ > 0 is the discrete distribution with pmf f ( x | λ ) = e- λ · λ x x ! , x = 0 , 1 , 2 , . . . Suppose a sample vector X = ( X 1 , . . ., X n ) is drawn from a Poisson distribution with unknown λ . Let W = ∑ n i =1 X i be the sample sum. 1. Write the pmf of the sample sum. SOLUTION: f W ( w | λ ) = e- nλ · ( nλ ) w w ! . 2. Write the likelihood function of λ , given that the sample is (2 , 3 , 1 , 2). 3. What is the MLE of λ in this case? 4. Show that in general, the MLE of λ is the sample mean. SOLUTION: L ( λ | vectorx ) = n productdisplay i =1 e- λ · λ x i x i ! ln L ( λ | vectorx ) =- nλ + ln λ n summationdisplay i =1 x i- summationdisplay ln( x i !) d dλ ln L ( λ | vectorx ) =- n + w λ , with w = ∑ n i =1 x i Thus, as a function of λ , L ( λ | vectorx ) is increasing on (0 , w/n ) and decreasing on ( w/n, ∞ )....
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