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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.

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Try: G. A. Whitmore & V. Seshadri (1987) A Heuristic Derivation of the Inverse Gaussian Distribution, The American Statistician, 41:4, 280-281

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