Central Limit Theorem

The central limit theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed random variables approaches a normal distribution, regardless of the original distribution of the variables: \[ \frac{\sum_{i=1}^{n} X_i - n\mu}{\sqrt{n\sigma^2}} \rightarrow N(0,1) \] where \( X_i \) are the random variables, \( \mu \) is the mean, \( \sigma^2 \) is the variance, and \( N(0,1) \) is the standard normal distribution.