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Maximum Likelihood Estimation (MLE) is one of the most important methods of statistical estimation used for determining the parameters of a probability distribution based on observed sample data. It provides a systematic approach for selecting the values of unknown parameters that make the observed data most probable. In Mathematical Foundations, MLE plays a significant role in estimation theory, probability models, and statistical inference. The basic principle of Maximum Likelihood Estimation is that the best estimate of an unknown parameter is the value that maximizes the likelihood of obtaining the observed sample. Suppose a random sample (X_1, X_2, ..., X_n) is drawn from a population with a probability distribution containing an unknown parameter (\theta). The ____ ___ ____ ______ _________ __________ _________ ____ _______ ______ ____ ____.
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