$\frac{S_n - n\mu}{\sigma\sqrt{n}} \xrightarrow{d} N(0,1) \quad \text{as } n \to \infty$ $\text{where } S_n = X_1 + \cdots + X_n$ The Central Limit Theorem (CLT) states that the standardized sum of independent and identically distributed random variables with mean $\mu$ and finite variance $\sigma^2$ converges in distribution to a standard normal as $n$ goes to infinity. For large $n$, this means $S_n$ is approximately $N(n\mu,\, n\sigma^2)$ regardless of the original distribution. > [!example]- Approximating Total Claims Across 200 Policies {💡 Example} > An insurer has 200 independent policies. Each policy's annual claim has mean \$500 and standard deviation \$300. Approximate $P(S_{200} > 104{,}000)$. > > > [!answer]- Answer > > By the CLT, $S_{200} \approx N(n\mu,\, n\sigma^2)$: > > $\mu_S = 200 \times 500 = 100{,}000, \qquad \sigma_S = \sqrt{200 \times 300^2} = 300\sqrt{200} \approx 4{,}243$ > > Standardising: > > $P(S_{200} > 104{,}000) = P\!\left(Z > \frac{104{,}000 - 100{,}000}{4{,}243}\right) = P(Z > 0.94) \approx 0.1736$