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Inverse Gaussian distribution
From Wikipedia, the free encyclopedia
Probability density function |
|
| Cumulative distribution function |
|
| Parameters | λ > 0 μ > 0 |
|---|---|
| Support |  |
| Probability density function (pdf) | ![\left[\frac{\lambda}{2 \pi x^3}\right]^{1/2} \exp{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}}](../../../math/2/c/6/2c6c7ef1a504f4998fe143af8151a018.png) |
| Cumulative distribution function (cdf) |  
where |
| Mean | μ |
| Median | |
| Mode | ![\mu\left[\left(1+\frac{9 \mu^2}{4 \lambda^2}\right)^\frac{1}{2}-\frac{3 \mu}{2 \lambda}\right]](../../../math/d/5/0/d50e806b11641fa78d3ddc5f7cbc0c0b.png) |
| Variance |  |
| Skewness |  |
| Excess kurtosis |  |
| Entropy | |
| Moment-generating function (mgf) | ![e^{\left(\frac{\lambda}{\mu}\right)\left[1-\sqrt{1-\frac{2\mu^2t}{\lambda}}\right]}](../../../math/d/f/b/dfb098ca17eedc92acafed44a7d67d8e.png) |
| Characteristic function | ![e^{\left(\frac{\lambda}{\mu}\right)\left[1-\sqrt{1-\frac{2\mu^2\mathrm{i}t}{\lambda}}\right]}](../../../math/6/b/b/6bbfc39a08f359b23390648366ec800d.png) |
is the The probability density function of the inverse Gaussian distribution is given by
![f(x;\mu,\lambda) = \left[\frac{\lambda}{2 \pi x^3}\right]^{1/2} \exp{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}}\mbox{ for } x > 0.](../../../math/3/5/3/353e4066a9bf3ce701952bfc989c8718.png)
The Wald distribution is simply another name for the inverse Gaussian distribution.
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 the distribution of distance at fixed time in Brownian motion, the inverse Gaussian describes the distribution of the time taken to reach a fixed distance.
The inverse of its cumulant generating function (logarithm of the characteristic function) is the inverse of the cumulant generating function of a Gaussian random variable.
- The inverse gaussian distribution: theory, methodology, and applications by Raj Chhikara and Leroy Folks, 1989 ISBN 0-8247-7997-5
- System Reliability Theory by Marvin Rausand and Arnljot Høyland
- Inverse Gaussian Distribution in Wolfram website.
