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The maximum likelihood estimator (MLE) is the value of the parameter that maximizes the likelihood function for the observed data. Graphically, it is the parameter value at the peak of the likelihood curve.

Conditions: A likelihood function $L(\theta \mid x_1, \dots, x_n)$ has been constructed.; One compares values of the parameter with the data fixed.

The maximum likelihood estimator (MLE) is the value of the parameter that maximizes the likelihood function for the observed data. Graphically, it is the parameter value at the peak of the likelihood curve.

Conditions: A likelihood function $L(\theta \mid x_1, \dots, x_n)$ has been constructed.; One compares values of the parameter with the data fixed.

To compute the likelihood for an independent sample, first write the joint pmf/pdf of all observations conditional on the parameter, then use independence to factor it into a product of the individual pmf/pdf terms.

Conditions: The sample $x_1, \dots, x_n$ is independent.; Each observation has the same functional form $f(x_i \mid \theta)$.; The data values are plugged in while $\theta$ remains symbolic.

To compute the likelihood for an independent sample, first write the joint pmf/pdf of all observations conditional on the parameter, then use independence to factor it into a product of the individual pmf/pdf terms.

Conditions: The sample $x_1, \dots, x_n$ is independent.; Each observation has the same functional form $f(x_i \mid \theta)$.; The data values are plugged in while $\theta$ remains symbolic.