The Central Limit Theorem

The Central Limit Theorem

The Central Limit Theorem is a fundamental result in probability theory that states that under certain conditions, the sum of a large number of independent random variables will have a distribution that approaches a normal distribution. In this proof, we will provide an intuitive and easy-to-follow explanation of the Central Limit Theorem.

Consider a large number of independent random variables X1, X2, ..., Xn with the same distribution and mean μ, and standard deviation σ. We want to find the distribution of the sum S = X1 + X2 + ... + Xn.

According to the Law of Large Numbers, the sample mean X̄ = (X1 + X2 + ... + Xn)/n approaches the true mean μ as n approaches infinity. That is, for any ε > 0, the probability that |X̄ - μ| > ε approaches 0 as n → ∞.

Now, we can apply the Chebyshev's inequality to the sum S, which states that for any ε > 0,

P(|S - μn| > ε) ≤ Var(S)/ε^2,

where μn = E(S) = nμ and Var(S) = nσ^2.

By substituting these expressions, we obtain:

P(|S - nμ| > ε) ≤ σ^2/(nε^2)

As n → ∞, the right-hand side of the inequality approaches 0, which means that the probability of the sum S deviating from nμ by more than ε becomes arbitrarily small. In other words, S converges in distribution to a normal distribution with mean nμ and variance nσ^2, as n approaches infinity.

This result has important practical implications, as it shows that even if the individual random variables do not follow a normal distribution, the sum of a large number of such variables will have a distribution that is approximately normal. This is why the normal distribution is ubiquitous in statistical analysis and is often used as an approximation for other distributions.

In conclusion, the Central Limit Theorem is a powerful result that helps us understand the behavior of sums of independent random variables. Our proof is intuitive and easy to follow, making it accessible to a wide audience.