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

Famous quotes containing the words inverse and/or distribution:

    “The quality of moral behaviour varies in inverse ratio to the number of human beings involved.”
    —Aldous Huxley (1894–1963)

    “There is the illusion of time, which is very deep; who has disposed of it? Mor come to the conviction that what seems the succession of thought is only the distribution of wholes into causal series.”
    —Ralph Waldo Emerson (1803–1882)