Stirling's Formula and Its Extensions: Heuristic Approaches

Abstract

Walsh (1995) introduced a heuristic approach to motivate Stirling's formula by equating a Poisson probability to an analogous value from a normal density function. We explore similar heuristics to derive approximations for various binomial, negative binomial, and multinomial coefficients. Also, using heuristics markedly different from those of Walsh, we develop an approximation of (nk)! for positive integers n (large) and k. These heuristics are then used to validate Stirling's formula for Γ(nα) where α is a positive real number. To derive each of our approximations we use a different probability distribution, and hence each section may serve as pedagogical module.

Keywords:

• Accuracy of approximations
• Binomial coefficient
• Factorial
• Gamma function
• Multinomial coefficient
• Negative binomial coefficient
• Normal approximation
• Stirling's approximation

Mathematics Subject Classification:

• Primary 62E17, 65C50
• Secondary 65C60

Acknowledgment

We thank the reviewers for helpful and encouraging comments.