Inverse gaussian distribution and its application

The inverse Gaussian distribution is related closely to the Gaussian distribution as is suggested by its name. It is used not only in mathematical statistics but also in various fields such as engineering to describe various phenomena and to make quantitative analysis. Schrodinger derived the cumulant generating function for the first-passage time of the Brownian motion with a drift. Then Tweedie extended Schrodinger's result and found that the cumulant-generating function for the time to travel for a unit distance and the cumulant-generating function for the distance traveled in the unit time are in the relation of the inverse function. Tweedie found also that the same relations exist between the binomial distribution and the negative binomial distribution as well as between Poisson distribution and gamma distribution. He called one of those the inverse of the other. The study of the inverse Gaussian distribution was developed further by Wasan, Johnson and Kotz; Folks and Chhikara; et al. This paper discusses the first-passage time distribution of the Wiener process as the origin of the inverse Gaussian distribution, the inverse Gaussian distribution itself and its properties, the generalized inverse distribution, and the application of the inverse Gaussian distribution.

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