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a method of estimating parameters of a statistical model by choosing the parameter value that optimizes the probability of observing the given sample.
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Maximum-likelihood estimator for data points with errors
As you have pointed out correctly, for data without errors Likelihood is $$ L(\theta) = \Pi_{i=1}^{N} f(x_i|\theta) = \Pi_{i=1}^{N}\frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x_i - \mu)^2}{\sigma^2}}$$ …