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| {{Probability distribution |
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| name =Inverse Gaussian|
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| type =density|
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| pdf_image =[[Image:PDF invGauss.svg|325px]]||
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| cdf_image =|
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| parameters =<math>\lambda > 0 </math> <br/><math> \mu > 0</math>|
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| support =<math> x \in (0,\infty)</math>|
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| pdf =<math> \left[\frac{\lambda}{2 \pi x^3}\right]^{1/2} \exp{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}}</math>|
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| 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>
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| where <math> \Phi \left(\right)</math> is the [[normal distribution|standard normal (standard Gaussian) distribution]] c.d.f. |
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| mean =<math> \mu </math>|
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| median =|
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| mode =<math>\mu\left[\left(1+\frac{9 \mu^2}{4 \lambda^2}\right)^\frac{1}{2}-\frac{3 \mu}{2 \lambda}\right]</math>|
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| variance =<math>\frac{\mu^3}{\lambda} </math>|
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| skewness =<math>3\left(\frac{\mu}{\lambda}\right)^{1/2} </math>|
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| kurtosis =<math>\frac{15 \mu}{\lambda} </math>|
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| entropy =|
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| mgf =<math>e^{\left(\frac{\lambda}{\mu}\right)\left[1-\sqrt{1-\frac{2\mu^2t}{\lambda}}\right]}</math>|
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| char =<math>e^{\left(\frac{\lambda}{\mu}\right)\left[1-\sqrt{1-\frac{2\mu^2\mathrm{i}t}{\lambda}}\right]}</math>|
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| }}
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| In [[probability theory]], the '''inverse Gaussian distribution''' (also known as the '''Wald distribution''') is a two-parameter family of [[continuous probability distribution]]s with [[support (mathematics)|support]] on (0,∞).
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| Its [[probability density function]] is given by
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| : <math> f(x;\mu,\lambda)
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| = \left[\frac{\lambda}{2 \pi x^3}\right]^{1/2} \exp{\frac{-\lambda (x-\mu)^2}{2 \mu^2 x}}</math>
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| for ''x'' > 0, where <math>\mu > 0</math> is the mean and <math>\lambda > 0</math> is the shape parameter.
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| As λ tends to infinity, the inverse Gaussian distribution becomes more like a [[normal distribution|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 [[Wiener process|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.
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| Its cumulant generating function (logarithm of the characteristic function) is the inverse of the cumulant generating function of a Gaussian random variable.
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| To indicate that a [[random variable]] ''X'' is inverse Gaussian-distributed with mean μ and shape parameter λ we write
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| :<math>X \sim IG(\mu, \lambda).\,\!</math> | |
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| ==Properties==
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| ===Summation===
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| If ''X''<sub>''i''</sub> has a IG(''μ''<sub>0</sub>''w''<sub>''i''</sub>, ''λ''<sub>0</sub>''w''<sub>''i''</sub><sup>2</sup>) distribution for ''i'' = 1, 2, ..., ''n''
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| and all ''X''<sub>''i''</sub> are [[statistical independence|independent]], then
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| :<math> | |
| S=\sum_{i=1}^n X_i
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| \sim
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| IG \left( \mu_0 \sum w_i, \lambda_0 \left(\sum w_i \right)^2 \right). </math>
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| Note that
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| :<math>
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| \frac{\textrm{Var}(X_i)}{\textrm{E}(X_i)}= \frac{\mu_0^2 w_i^2 }{\lambda_0 w_i^2 }=\frac{\mu_0^2}{\lambda_0}
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| </math>
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| is constant for all ''i''. This is a [[Necessary and sufficient conditions|necessary condition]] for the summation. Otherwise ''S'' would not be inverse Gaussian.
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| ===Scaling===
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| For any ''t'' > 0 it holds that
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| :<math>
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| X \sim IG(\mu,\lambda) \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\, tX \sim IG(t\mu,t\lambda).
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| </math>
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| ===Exponential family===
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| The inverse Gaussian distribution is a two-parameter [[exponential family]] with [[natural parameters]] -λ/(2μ²) and -λ/2, and [[natural statistics]] ''X'' and ''1/X''.
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| ==Relationship with Brownian motion==
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| The [[stochastic process]] ''X''<sub>''t''</sub> given by
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| :<math>X_0 = 0\quad</math>
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| :<math>X_t = \nu t + \sigma W_t\quad\quad\quad\quad</math>
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| (where ''W''<sub>''t''</sub> is a standard [[Wiener process|Brownian motion]] and <math>\nu > 0</math>)
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| is a Brownian motion with drift ''ν''.
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| Then, the [[first passage time]] for a fixed level <math>\alpha > 0</math> by ''X''<sub>''t''</sub> is distributed according to an inverse-gaussian:
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| :<math>T_\alpha = \inf\{ 0 < t \mid X_t=\alpha \} \sim IG(\tfrac\alpha\nu, \tfrac {\alpha^2} {\sigma^2}).\,</math>
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| ===When drift is zero===
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| A common special case of the above arises when the Brownian motion has no drift. In that case, parameter
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| μ tends to infinity, and the first passage time for fixed level α has probability density function
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| : <math> f \left( x; 0, \left(\frac{\alpha}{\sigma}\right)^2 \right)
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| = \frac{\alpha}{\sigma \sqrt{2 \pi x^3}} \exp\left(-\frac{\alpha^2 }{2 x \sigma^2}\right).</math>
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| This is a [[Lévy distribution]] with parameters <math>c=\frac{\alpha^2}{\sigma^2}</math> and <math>\mu=0</math>.
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| ==Maximum likelihood==
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| The model where
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| :<math>
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| X_i \sim IG(\mu,\lambda w_i), \,\,\,\,\,\, i=1,2,\ldots,n
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| </math>
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| with all ''w''<sub>''i''</sub> known, (''μ'', ''λ'') unknown and all ''X''<sub>''i''</sub> [[statistical independence|independent]] has the following likelihood function
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| :<math>
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| L(\mu, \lambda)=
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| \left( \frac{\lambda}{2\pi} \right)^\frac n 2
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| \left( \prod^n_{i=1} \frac{w_i}{X_i^3} \right)^{\frac{1}{2}}
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| \exp\left(\frac{\lambda}{\mu}\sum_{i=1}^n w_i -\frac{\lambda}{2\mu^2}\sum_{i=1}^n w_i X_i - \frac\lambda 2 \sum_{i=1}^n w_i \frac1{X_i} \right).
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| </math>
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| Solving the likelihood equation yields the following maximum likelihood estimates
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| :<math>
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| \hat{\mu}= \frac{\sum_{i=1}^n w_i X_i}{\sum_{i=1}^n w_i}, \,\,\,\,\,\,\,\, \frac{1}{\hat{\lambda}}= \frac{1}{n} \sum_{i=1}^n w_i \left( \frac{1}{X_i}-\frac{1}{\hat{\mu}} \right).
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| </math>
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| <math>\hat{\mu}</math> and <math>\hat{\lambda}</math> are independent and
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| :<math>
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| \hat{\mu} \sim IG \left(\mu, \lambda \sum_{i=1}^n w_i \right) \,\,\,\,\,\,\,\, \frac{n}{\hat{\lambda}} \sim \frac{1}{\lambda} \chi^2_{n-1}.
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| </math>
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| ==Generating random variates from an inverse-Gaussian distribution==
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| The following algorithm may be used.<ref>''Generating Random Variates Using Transformations with Multiple Roots'' by John R. Michael, William R. Schucany and Roy W. Haas, ''[[American Statistician]]'', Vol. 30, No. 2 (May, 1976), pp. 88–90</ref>
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| Generate a random variate from a normal distribution with a mean of 0 and 1 standard deviation
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| :<math>
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| \displaystyle \nu = N(0,1).
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| </math>
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| Square the value
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| :<math>
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| \displaystyle y = \nu^2
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| </math>
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| and use this relation
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| :<math>
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| x = \mu + \frac{\mu^2 y}{2\lambda} - \frac{\mu}{2\lambda}\sqrt{4\mu \lambda y + \mu^2 y^2}.
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| </math>
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| Generate another random variate, this time sampled from a uniform distribution between 0 and 1
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| :<math>
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| \displaystyle z = U(0,1).
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| </math>
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| If
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| :<math>
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| z \le \frac{\mu}{\mu+x}
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| </math>
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| then return
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| :<math>
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| \displaystyle
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| x
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| </math>
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| else return
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| :<math>
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| \frac{\mu^2}{x}.
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| </math>
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| Sample code in [[Java (programming language)|Java]]:
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| <source lang="java">
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| public double inverseGaussian(double mu, double lambda) {
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| Random rand = new Random();
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| double v = rand.nextGaussian(); // sample from a normal distribution with a mean of 0 and 1 standard deviation
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| double y = v*v;
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| double x = mu + (mu*mu*y)/(2*lambda) - (mu/(2*lambda)) * Math.sqrt(4*mu*lambda*y + mu*mu*y*y);
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| double test = rand.nextDouble(); // sample from a uniform distribution between 0 and 1
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| if (test <= (mu)/(mu + x))
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| return x;
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| else
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| return (mu*mu)/x;
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| }
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| </source>
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| [[File:Wald Distribution matplotlib.jpg|thumb|right|Wald Distribution using Python with aid of matplotlib and NumPy]]
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| And to plot Wald distribution in [[Python (programming language)|Python]] using [[matplotlib]] and [[Numpy|NumPy]]:
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| <source lang="python">
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| import matplotlib.pyplot as plt
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| import numpy as np
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| h = plt.hist(np.random.wald(3, 2, 100000), bins=200, normed=True)
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| plt.show()
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| </source>
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| ==Related distributions==
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| * If <math> X \sim \textrm{IG}(\mu,\lambda)\,</math> then <math> k X \sim \textrm{IG}(k \mu,k \lambda)\,</math>
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| * If <math> X_i \sim \textrm{IG}(\mu,\lambda)\,</math> then <math> \sum_{i=1}^{n} X_i \sim \textrm{IG}(n \mu,n^2 \lambda)\,</math>
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| * If <math> X_i \sim \textrm{IG}(\mu,\lambda)\,</math> for <math>i=1,\ldots,n\,</math> then <math> \bar{X} \sim \textrm{IG}(\mu,n \lambda)\,</math>
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| * If <math> X_i \sim \textrm{IG}(\mu_i,2 \mu^2_i)\,</math> then <math> \sum_{i=1}^{n} X_i \sim \textrm{IG}\left(\sum_{i=1}^n \mu_i, 2 {\left( \sum_{i=1}^{n} \mu_i \right)}^2\right)\,</math>
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| The convolution of a Wald distribution and an exponential (the ex-Wald distribution) is used as a model for response times in psychology.<ref name=Schwarz2001>Schwarz W (2001) The ex-Wald distribution as a descriptive model of response times. Behav Res Methods Instrum Comput 33(4):457-469</ref>
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| ==History==
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| This distribution appears to have been first derived by Schrödinger in 1915 as the time to first passage of a Brownian motion.<ref name=Schrodinger1915>Schrodinger E (1915) Zur Theorie der Fall—und Steigversuche an Teilchenn mit Brownscher Bewegung. Physikalische Zeitschrift 16, 289-295</ref> The name inverse Gaussian was proposed by Tweedie in 1945.<ref name=Folks1978>Folks JL & Chhikara RS (1978) The inverse Gaussian and its statistical application - a review. J Roy Stat Soc 40(3) 263-289</ref> Wald re-derived this distribution in 1947 as the limiting form of a sample in a sequential probability ratio test. Tweedie investigated this distribution in 1957 and established some of its statistical properties.
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| ==Software==
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| The R programming language has software for this distribution. [http://www.stat.ucl.ac.be/ISdidactique/Rhelp/library/SuppDists/html/invGauss.html]
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| == See also == | |
| *[[Generalized inverse Gaussian distribution]]
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| * [[Tweedie distributions]]—The inverse Gaussian distribution is a member of the family of Tweedie [[exponential dispersion model]]s
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| *[[Stopping time]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * ''The inverse gaussian distribution: theory, methodology, and applications'' by Raj Chhikara and Leroy Folks, 1989 ISBN 0-8247-7997-5
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| * ''System Reliability Theory'' by Marvin Rausand and [[Arnljot Høyland]]
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| * ''The Inverse Gaussian Distribution'' by Dr. V. Seshadri, Oxford Univ Press, 1993
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| ==External links==
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| * [http://mathworld.wolfram.com/InverseGaussianDistribution.html Inverse Gaussian Distribution] in Wolfram website.
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{DEFAULTSORT:Inverse Gaussian Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Exponential family distributions]]
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| [[Category:Probability distributions]]
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