How to Solve Sampling and Limit Theorems — Normal distribution — CLT approximation of a total Questions on Exam P

The total cost is a sum of many independent, identically distributed exponential costs. The central limit theorem approximates it by a normal variable with additive mean and variance, allowing the loading margin to be standardized. Let \(S=\sum_{i=1}^{100} X_i\) be total cost. Each \(X_i\sim\operatorname{Exp}(1/1000)\) has mean \(1000\) and variance \(1000^2\). By the Central Limit Theorem, \(S\) is approximately normal with mean \(\mu_S=100(1000)=100000\) and standard deviation \(\sigma_S=1000\sqrt{100}=10000\). Collected fees equal \(100(1000)\left(1+\dfrac{5}{100}\right)=100000+1000(5)\). Standardizing fees gives \(z=\dfrac{(100000+1000(5))-100000}{10000}=0.5\). The probability that total cost exceeds collected fees is upper-tail \(\Pr(S>\text{fees})\approx 1-\Phi(\frac{1}{2})=0.3085\). Thus \(\Pr(\text{total cost exceeds fees})=0.3085\approx0.3085\), which matches the correct option.