# 10_05 - STAT 410 Examples for Fall 2011 p.m.f or p.d.f f x...

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Examples for 10/05/2011 Fall 2011 p.m.f. or p.d.f. f ( x ; θ ) , θ Ω . Ω – parameter space. Likelihood function: L ( θ ) = L ( θ ; x 1 , x 2 , … , x n ) = = n i 1 f ( x i ; θ ) = f ( x 1 ; θ ) f ( x n ; θ ) It is often easier to consider ln L ( θ ) = = n i 1 ln f ( x i ; θ ) . 1. Let X 1 , X 2 , … , X n be a random sample of size n from a Poisson distribution with mean λ , λ > 0. a) Obtain the maximum likelihood estimator of λ , λ ˆ . ( ) ( ) = - = = = n i i n i i e i f 1 X 1 X λ ; X L ! λ λ λ . ( ) ( ) = = - - = n i i n i i n 1 1 ! λ λ λ X ln ln X L ln . ( ) n n i i d d - = = 1 X 1 L ln λ λ λ = 0. X X 1 λ ˆ = = = n n i i . Method of Moments: E ( X ) = g ( θ ). Set X = g ( θ ~ ). Solve for θ ~ . b) Obtain the method of moments estimator of λ , λ ~ . E

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10_05 - STAT 410 Examples for Fall 2011 p.m.f or p.d.f f x...

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