Equating Poisson and Normal Probability Functions to Derive Stirling's Formula

Abstract

Often texts for introductory courses in probability or mathematical statistics make reference to Stirling's asymptotic formula (for a factorial) without presenting any proof or justification for the formula. A notable exception is Feller. In this article, we present a derivation of Stirling's formula based on the normal approximation of a Poisson probability that is considerably more accessible to the average student than Feller's approach. Besides illustrating a usage of the central limit theorem in conjunction with a continuity correction, the derivation lends itself to a mnemonic device for quickly obtaining Stirling's formula.

Key Words:

• Central limit theorem
• Continuity correction