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The Central Limit Theorem (CLT) is one of the most fundamental and powerful results in probability theory and statistics. It explains the behavior of the sampling distribution of the sample mean when random samples are drawn from a population. The theorem states that if a sufficiently large number of independent and identically distributed random samples are taken from any population with a finite mean and finite variance, then the distribution of the sample mean approaches a normal distribution, regardless of the shape of the original population distribution. This remarkable property makes the normal distribution central to statistical analysis and inference. Mathematically, suppose a population has mean ( \mu ) and variance ( \sigma^2 ). If random samples of size ( n ) are selected independently, then the sampling distribution of the sample mean ( \bar{X} ) has mean ( \mu ____ _______ _________ _________ _______ _______ _________ ___ _______ ______.
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