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Maximum Likelihood Estimation of Normal Distribution

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The probability density (likelihood) of a single variable drawn from a Normal distribution  is:

 

 

Thus the likelihood of  independent draws  from this distribution is:

 

 

To identify the values of  and  that maximise  it is easiest to identify the maximum of the log likelihood (as the two will be maximised for the same values of  and  since  is a monotonically increasing function of ). The log likelihood is:

 

 

This is maximised when  and , i.e. when:

 

 

The maximum likelihood estimator, , of the mean  is thus the average of the observations, . It is possible to show that this is also the minimum variance unbiased estimator of . The maximum likelihood estimator, , of  is the population standard deviation,  of the  which can be determined using the MnPopulationStdev web function. Please note that whilst  is an unbiased estimator of ,  is a biased estimator of . The minimum variance unbiased estimator of  is the sample variance (i.e. square of the sample standard deviation). The sample standard deviation is:

 

 


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