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Likelihood Questions - Example 10:15 Given x successes in n...

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Example 10 : 15 . Given x °successes±²in n trials, ³nd the maximum likeli- hood estimate of the parameter ° of the corresponding binomial distribution. Solution 10 : 15 . To ³nd the value of ° that maximises L ( ° ) = ° n x ± ° x (1 ° ° ) n ° x it will be convenient to make use of the fact that the value of ° that maximises L ( ° ) will also maximise ln L ( ° ) = ln ° n x ± + x ln ° + ( n ° x ) ln(1 ° ° ) : Thus, we get d (ln L ( ° )) = x ° ° n ° x 1 ° ° ; and, equating this derivative to 0 and solving for °; we ³nd that the likelihood function has a maximum at ° = x n : This is the maximum likelihood estimate of the binomial parameter ° , and we refer to ^ ° = X n as the corresponding maximum likelihood estimator. Example 10 : 16 . If x 1 ; x 2 ; :::; x n are the values of a random sample from an
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