Inverse Gaussian distribution
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| Probability density function Image:InverseG.png | |
| Cumulative distribution function | |
| Parameters | <math>\lambda > 0 </math> <math> \mu > 0</math> |
|---|---|
| Support | <math> x \in (0,\infty)</math> |
| Probability density function (pdf) | <math> \left[\frac{\lambda}{2 \pi x^3}\right]^{1/2} \exp{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x |
| Cumulative distribution function (cdf) | {{{cdf}}} |
| Mean | {{{mean}}} |
| Median | {{{median}}} |
| Mode | {{{mode}}} |
| Variance | {{{variance}}} |
| Skewness | {{{skewness}}} |
| Excess Kurtosis | {{{kurtosis}}} |
| Entropy | {{{entropy}}} |
| mgf | {{{mgf}}} |
| Char. func. | {{{char}}} |
cdf =<math> \Phi\left(\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu}-1 \right)\right) </math> <math>+\exp\left(\frac{2 \lambda}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}+1 \right)\right) </math>
where <math> \Phi \left(\right)</math> is the normal (Gaussian) distribution c.d.f. |
mean =<math> \mu </math>|
median =|
mode =<math>\mu\left[\left(1+\frac{9 \mu^2}{4 \lambda^2}\right)^\frac{1}{2}-\frac{3 \mu}{2 \lambda}\right]</math>|
variance =<math>\frac{\mu^3}{\lambda} </math>|
skewness =<math>3\left(\frac{\mu}{\lambda}\right)^{1/2} </math>|
kurtosis =<math>3 +\frac{15 \mu}{\lambda} </math>|
entropy =|
mgf =<math>e^{\left(\frac{\lambda}{\mu}\right)\left[1-\sqrt{1-\frac{2\mu^2x}{\lambda}}\right]}</math>|
char =|
}}
The probability density function of the inverse Gaussian distribution is given by
- <math>
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>
The Wald distribution is the special case of the inverse Gaussian distribution in which μ = λ = 1.
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.
[edit] References
- The Inverse Gaussian distribution by Raj Chhikara and Leroy Folks
- System Reliability Theory by Marvin Rausand and Arnljot Høyland
[edit] External link
- Inverse Gaussian Distribution in Wolfram website.
