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DISTRIBUTIONS / CONTINUOUS / INVERSE GAUSSIAN

Inverse Gaussian distribution

What it describes

The Inverse Gaussian distribution models a first-passage time for Brownian motion with positive drift. Its historical name is misleading: it is not the reciprocal of an ordinary normal variable.

It is positive. mu controls mean lifetime and lambda concentration around it; the right tail represents paths that take a long time to hit the threshold.

History and names

Schrödinger connected the law to first-passage times in 1915, and Wald made it visible in sequential analysis. The Wiener-process connection is more informative than the word “inverse”.

How it connects to other distributions

It is linked to Fatigue Life/Birnbaum–Saunders in fatigue models and to Gamma in some mixtures. Its first-passage representation distinguishes it from purely phenomenological lognormal and Weibull models.

Where it appears

  • reaction times, survival, and first-arrival times
  • hydrology, reliability, and threshold-accumulation models

Fitting cautions

Do not read lambda as a Poisson rate or copy a formula from another scale convention without translating it.

The time a wandering path takes to reach a barrier

Imagine a particle moving with positive drift while Brownian motion pushes it off course. The first time it reaches a barrier has an Inverse Gaussian distribution. The variable is not an inverted Normal: the adjective comes from historical relationships among generating functions.

Schrödinger studied this law in 1915, and Wald made it central to sequential analysis. It is consequently also called the Wald distribution. Its first-passage interpretation separates it from Gamma and Weibull when a plausible cumulative path exists. With weak drift, its right tail can become pronounced. If the observations are merely independent durations with no barrier or accumulation, a good visual fit does not establish the Brownian story.

Decision guide

A good candidate when: the variable is a positive first-passage time or a duration whose skewness decreases with the mean-to-shape ratio.

Compare it with: Gamma and Lognormal. Compare not only density but also the mean–variance relationship and the barrier-crossing mechanism.

References

  • SciPy reference: scipy.stats.invgauss — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Schrödinger, E. (1915). Zur Theorie der Fall- und Steigversuche an Teilchen mit Brownscher Bewegung. Physikalische Zeitschrift, 16, 289–295.
  • Wald, A. (1947). Sequential Analysis. Wiley.

Inverse Gaussian Distribution: equations and calculator

Distribution defintion

X∼InverseGaussian(μ,λ)X\sim\mathrm{InverseGaussian}\left(\mu,\lambda\right)

Distribution domain

x∈(0,∞)x\in\left(0,\infty\right)

Parameters domain and parameters constraints

μ∈R+,λ∈R+\mu\in\mathbb{R}^{+},\lambda\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=Φ(λx(xμ−1))+exp⁡(2λμ)Φ(−λx(xμ+1))F_{X}\left(x\right)=\Phi\left(\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}-1\right)\right)+\exp\left(\frac{2 \lambda}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}+1\right)\right)

Probability density function

fX(x)=λ2πx3exp⁡[−λ(x−μ)22μ2x]f_{X}\left(x\right)=\sqrt\frac{\lambda}{2 \pi x^3} \exp\left[-\frac{\lambda (x-\mu)^2}{2 \mu^2 x}\right]

Percent point function/Sample

SampleX={x0if u2⩽μμ+x0μ2x0if u2⩾μμ+x0\text{Sample}_{X}=\left\{\begin{array}{cl} x_{0} \quad \text{if } u_{2}\leqslant\frac{\mu}{\mu+x_{0}}\\ \frac{\mu^{2}}{x_{0}} \quad \text{if } u_{2}\geqslant \frac{\mu}{\mu+x_{0}} \end{array} \right.

Non-central parametric moments

μk′=E[Xk]=∫0∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{0}^{\infty }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=μ\mathrm{Mean}(X)=\mu'_{1}=\mu

Parametric variance

Variance(X)=μ2′−μ1′2=μ3λ\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{\mu^3}{\lambda}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=3(μλ)1/2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=3\left(\frac{\mu}{\lambda}\right)^{1/2}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3+15μλ\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=3+\frac{15 \mu}{\lambda}

Parametric median

Median(X)=FX−1(12)\mathrm{Median}(X)=F^{-1}_{X}\left(\frac{1}{2}\right)

Parametric mode

Mode(X)=μ[(1+9μ24λ2)12−3μ2λ]\mathrm{Mode}(X)=\mu\left[\left(1+\frac{9 \mu^2}{4 \lambda^2}\right)^\frac{1}{2}-\frac{3 \mu}{2 \lambda}\right]

Additional information and definitions

Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution.
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}
x0=μ+μ2[Φ−1(u1)]22λ−μ2λ4μλ[Φ−1(u1)]2+μ2([Φ−1(u1)]2)2x_{0}=\mu+\frac{\mu^2 [\Phi^{-1}\left(u_{1}\right)]^{2}}{2\lambda}-\frac{\mu}{2\lambda}\sqrt{4\mu \lambda [\Phi^{-1}\left(u_{1}\right)]^{2}+\mu^2 ([\Phi^{-1}\left(u_{1}\right)]^{2})^2}
u1:Uniform[0,1] random varibleu_{1}:\text{Uniform[0,1] random varible}
u2:Uniform[0,1] random varibleu_{2}:\text{Uniform[0,1] random varible}