Central Limit Theorem (CLT)

The Central Limit Theorem (CLT) states that the sum or average of a large number of independent, identically distributed random variables with finite mean $\mu$ and variance $\sigma^2$ is approximately normal, regardless of the original distribution.

$\frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0, 1)$

Equivalently, a sum $S_n = \sum_{i=1}^{n} X_i$ is approximately $N(n\mu,\ n\sigma^2)$ for large $n$ .
It justifies the normal approximation used throughout actuarial modeling of aggregate losses.
A continuity correction improves the approximation when the $X_i$ are discrete.

Normal Approximation of Aggregate Losses An insurer has 10,000 independent policies, each with mean loss $ $200$ and standard deviation $ $80$ . Approximate the probability that total losses exceed $ $2{,}050{,}000$ .
Answer Let $S = \sum_{i=1}^{10000} X_i$ . By the CLT, $S$ is approximately normal with:
 $\begin{align*} E[S] &= 10{,}000 \times 200 = 2{,}000{,}000 \\ \sigma_S &= 80\sqrt{10{,}000} = 8{,}000 \end{align*}$ 
Standardize the threshold: $Z = \frac{2{,}050{,}000 - 2{,}000{,}000}{8{,}000} = 6.25$ $P(S > 2{,}050{,}000) = P(Z > 6.25) \approx 0$ A total that far above the mean is essentially impossible.