08_26_09 - STAT 409 p.m.f or p.d.f 1 Fall 2009 Examples for...

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STAT 409 Examples for 08/26/2009 Fall 2009 p.m.f. or p.d.f. f ( x ; θ ) , θ . – parameter space. 1. Suppose = { 1, 2, 3 } and the p.m.f. f ( x ; θ ) is θ = 1: f ( 1 ; 1 ) = 0.6, f ( 2 ; 1 ) = 0.1, f ( 3 ; 1 ) = 0.1, f ( 4 ; 1 ) = 0.2. θ = 2: f ( 1 ; 2 ) = 0.2, f ( 2 ; 2 ) = 0.3, f ( 3 ; 2 ) = 0.3, f ( 4 ; 2 ) = 0.2. θ = 3: f ( 1 ; 3 ) = 0.3, f ( 2 ; 3 ) = 0.4, f ( 3 ; 3 ) = 0.2, f ( 4 ; 3 ) = 0.1. What is the maximum likelihood estimate of θ ( based on only one observation of X ) if … a) X = 1; b) X = 2; c) X = 3; d) X = 4. 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 ) . 2. If the random variable Y denotes an individual’s income, Pareto’s law claims that P ( Y y ) = θ y k , where k is the entire population’s minimum income. It follows that f Y ( y ) = 1
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08_26_09 - STAT 409 p.m.f or p.d.f 1 Fall 2009 Examples for...

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