Central Limit Theorem

Let \( X_1, X_2, \dots, X_n \) be a sample of independent and identically distributed random variables with mean \( \mu \) and variance \( \sigma^2 \).

Then, the sample mean \( \bar{X} = \frac{1}{n}\sum_{i=1}^{n}X_i \) approaches a normal distribution with mean \( \mu \) and variance \( \frac{\sigma^2}{n} \) as \( n \) tends to infinity.