The Central Limit Theorem

The law of large numbers says $M_n\to\mu$, but says nothing about the shape of the (shrinking) fluctuations around $\mu$ along the way. The Central Limit Theorem (CLT) answers exactly that question, and the answer is the single most important reason the normal distribution is ubiquitous in probabilistic modelling: at the natural $1/\sqrt n$ scale, the fluctuations of any sum of i.i.d. random variables — discrete, continuous, or mixed, whatever their own shape — converge to a normal distribution. This note states the theorem, develops the standard normal-approximation procedure, and works three examples that show the CLT in action: a package-weight sum, a counting problem disguised as a sum, and — most tellingly — a return to the polling problem of the previous two lessons, where the CLT's sample-size requirement turns out to be more than five times sharper than Chebyshev's.

Learning objectives. By the end of this note you should be able to: