Inverse Gaussian Distribution

In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,∞).

Its probability density function is given by

f(x;\mu,\lambda) = \left^{1/2} \exp{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}}

for x > 0, where is the mean and is the shape parameter.

As λ tends to infinity, the inverse Gaussian distribution becomes more like a normal (Gaussian) distribution. The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading: it is an "inverse" only in that, while the Gaussian describes a Brownian Motion's level at a fixed time, the inverse Gaussian describes the distribution of the time a Brownian Motion with positive drift takes to reach a fixed positive level.

Its cumulant generating function (logarithm of the characteristic function) is the inverse of the cumulant generating function of a Gaussian random variable.

To indicate that a random variable X is inverse Gaussian-distributed with mean μ and shape parameter λ we write

Read more about Inverse Gaussian Distribution:  Relationship With Brownian Motion, Maximum Likelihood, Generating Random Variates From An Inverse-Gaussian Distribution, Related Distributions, History, Software

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