Chhikara and Folks (1988) show that the inverse gaussian distribution arises as the first passage time for a wiener process. However, there are several steps I don't quite understand. In particular, some of the steps, e.g. the last one on p. 26, in finding the Laplace transform $f^*$, are rather opaque.
Are there other places where this derivation is shown? Google yields nothing so far.
There are three approaches:
The heuristic derivation as the one by Whitmore and Seshadri described in the other answer.
A derivation based on solving the differential equations of a diffusion process with drift. You can read about it in (but apparently Erwin Schrödinger also described it, but the source is unavailable to me)
Smoluchowski, Marian V. "Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung." Annalen der Physik 353.24 (1916): 1103-1112. (online available via: https://www.physik.uni-augsburg.de/theo1/hanggi/History/BM-History.html)
Who was the first person to prove the straight line cross probability for a Brownian motion?
An inversion process described by Tweedie. (the cumulant generating function of the Gaussian and the inverse Gaussian are inversed)
Tweedie, Maurice CK. "Inverse statistical variates." Nature 155.3937 (1945): 453-453.
Tweedie, M. C. K. "Functions of a statistical variate with given means, with special reference to Laplacian distributions." Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 43. No. 1. Cambridge University Press, 1947.
This derivation, involving the Laplace transform as well, is probably the one that you are studying.
Try: G. A. Whitmore & V. Seshadri (1987) A Heuristic Derivation of the Inverse Gaussian Distribution, The American Statistician, 41:4, 280-281
