Central Limit Theorem

Study guides: Exam P-1, Exam MAS-I

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 $n \to \infty$ .

$\frac{S_n - n\mu}{\sigma\sqrt{n}} \xrightarrow{d} N(0,1) \quad \text{as } n \to \infty$

$\text{where } S_n = X_1 + \cdots + X_n$

For large $n$ , $S_n$ is approximately $N(n\mu,\, n\sigma^2)$ regardless of the original distribution

Approximating Total Claims Across 200 Policies 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 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: $P(S_{200} > 104{,}000) = P\!\left(Z > \frac{104{,}000 - 100{,}000}{4{,}243}\right) = P(Z > 0.94) \approx 0.1736$