Central Limit Theorem Proof

The Central Limit Theorem states that if you take many random samples from a population and calculate their averages, those averages will form a bell-shaped, normal distribution, regardless of the population's original shape. The proof involves mathematical reasoning showing that sums of independent, identically distributed variables tend to follow a normal distribution as sample size increases, thanks to properties of moments and characteristic functions. Essentially, it’s about how repeated averaging smooths out individual irregularities, making the distribution of sample means predictable and approximately normal when the sample size is large enough.