Central Limit Theorem

The Central Limit Theorem says that when you add many independent and identically distributed random variables, the standardized sum tends to look normal even if the underlying variables are not normal.

That does not mean everything is magically normal. It means that for large enough aggregation, the shape of the sum becomes much easier to approximate using the mean and variance alone.

Suppose daily claim counts are iid with mean 12 and variance 9. Over 36 days, the total count S has mean 36(12)=432 and variance 36(9)=324, so its standard deviation is 18.

To approximate P(S > 450), standardize: Z=(450-432)/18=1. The CLT then gives P(S > 450) approximately equal to P(Z > 1), which is about 0.159. The important step is setting up the mean and variance of the aggregate correctly before reaching for the normal table.

The July 2026 syllabus explicitly calls for CLT approximations for linear combinations of iid random variables. That makes the theorem a direct exam skill, not just a background idea.

It is also one of the cleanest bridges between the univariate and multivariate parts of Exam P. Once you know how moments behave under sums, the CLT tells you when a normal approximation is the practical next move.

A common mistake is using the original variance instead of the variance of the sum or average. Another is applying a normal approximation mechanically without checking whether the setup actually involves repeated iid variables or a large enough aggregation to make the approximation sensible.

The CLT is one of the main reasons sample means, regression summaries, and portfolio totals are often modeled with normal-based uncertainty. Exam P uses the theorem in a lighter form, but the modeling logic is exactly the same.