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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 = 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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- Fall '10