From the Poisson Distribution to Stirling's Approximation

The Poisson distribution is the most famous probability model for counts, non-negative integer values. Many real-world phenomena are well approximated by this distribution, including the number of German bombs that landed in 1/4km grid squares in south London during WWII.

Thirty isn't the magic number

The simplest version of the central limit theorem (CLT) says that if \(X_1, \dots, X_n\) are iid random variables with mean \(\mu\) and finite variance \(\sigma^2\) \[ \frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \rightarrow_d N(0,1) \] where \(\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i\).