TITLE:
Elementary Derivation of First-Passage Distribution
AUTHORS:
Jan Vrbik
KEYWORDS:
Brownian Motion, Random Walk, Gambler’s Ruin Problem, Stirling’s Formula, Probability Generating Function
JOURNAL NAME:
Open Journal of Statistics,
Vol.16 No.5,
September
29,
2026
ABSTRACT: One of the most intricate formulas relating to a Stochastic process known as Brownian motion specifies the probability density function of the time the process takes to reach, for the first time, a given state. There are several existing derivations thereof (a special case of inverse Gaussian distribution), most of them requiring sophisticated mathematical tools and rather complex reasoning. In this article, we present a simple derivation of the same result, based on elementary arguments and rudimentary mathematics only.