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Unformatted text preview: Estimation – More Examples Example 1 Let X 1 ;X 2 ;:::;X n be a random sample from the Poisson distribution with mean ¸ . a. Find a point estimator for ¸ using the …rst moment with the method of moments technique. Solution The …rst moment is: E ( X ) = ¸: By equating this to the …rst theoretical moment and solving for ¸ , we see that: ^ ¸ = P n i =1 x i n : b. Find the maximum likelihood estimator for ¸ . Solution The log-likelihood is: l ( ¸ ) = n X i =1 ( x i ln( ¸ ) ¡ ln( x !)) ¡ ¸n: The …rst derivative is: dl ( ¸ ) d¸ = Ã n X i =1 x i ¸ ! ¡ n: Solving dl ( ¸ ) d¸ = 0 for ¸ , we see that the maximum likelihood estimator is: ^ ¸ = P n i =1 x i n : Example 2 A random sample of n bike helmets manufactured by a certain company is selected. Let X be the number among the n that are ‡awed and let p be the probability that a randomly selected helmet is ‡awed. Assume that only X is observed (not the sequences of successes and failures)....
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This note was uploaded on 01/08/2012 for the course EXST 4050 taught by Professor Staff during the Fall '10 term at LSU.
- Fall '10