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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 distributions with support on (0,∞).

Its probability density function is given by

f ( x ; μ μ , λ λ ) = λ λ 2 π π x 3 exp ⁡ ⁡ ( − − λ λ ( x − − μ μ ) 2 2 μ μ 2 x ) {\displaystyle f(x;\mu ,\lambda )={\sqrt {\frac {\lambda }{2\pi x^{3}}}}\exp {\biggl (}-{\frac {\lambda (x-\mu )^{2}}{2\mu ^{2}x}}{\biggr )}}

for x > 0, where μ μ > 0 {\displaystyle \mu >0} is the mean and λ λ > 0 {\displaystyle \lambda >0} is the shape parameter.cite-ref-chhikara1989-1-0[1]

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 X ∼ ∼ IG ⁡ ⁡ ( μ μ , λ λ ) {\displaystyle X\sim \operatorname {IG} (\mu ,\lambda )\,\!} .

• Properties

• Single parameter form

• Summation

• Scaling

• Exponential family

• Relationship with Brownian motion

• When drift is zero

• Maximum likelihood

• Sampling from an inverse-Gaussian distribution

• Related distributions

• History

• Rated Inverse Gaussian Distribution

• Numeric computation and software

• See also

• References

• Further reading

• External links

The probability density function (pdf) of the inverse Gaussian distribution has a single parameter form given by

f ( x ; μ μ , μ μ 2 ) = μ μ 2 π π x 3 exp ⁡ ⁡ ( − − ( x − − μ μ ) 2 2 x ) . {\displaystyle f(x;\mu ,\mu ^{2})={\frac {\mu }{\sqrt {2\pi x^{3}}}}\exp {\biggl (}-{\frac {(x-\mu )^{2}}{2x}}{\biggr )}.}

In this form, the mean and variance of the distribution are equal, E [ X ] = Var ( X ) . {\displaystyle \mathbb {E} [X]={\text{Var}}(X).}

Also, the cumulative distribution function (cdf) of the single parameter inverse Gaussian distribution is related to the standard normal distribution by

Pr ( X < x ) = Φ Φ ( − − z 1 ) + e 2 μ μ Φ Φ ( − − z 2 ) , {\displaystyle {\begin{aligned}\Pr(X<x)&=\Phi (-z_{1})+e^{2\mu }\Phi (-z_{2}),\end{aligned}}}

where z 1 = μ μ x 1 / 2 − − x 1 / 2 {\displaystyle z_{1}={\frac {\mu }{x^{1/2}}}-x^{1/2}} , z 2 = μ μ x 1 / 2 + x 1 / 2 , {\displaystyle z_{2}={\frac {\mu }{x^{1/2}}}+x^{1/2},} and the Φ Φ {\displaystyle \Phi } is the cdf of standard normal distribution. The variables z 1 {\displaystyle z_{1}} and z 2 {\displaystyle z_{2}} are related to each other by the identity z 2 2 = z 1 2 + 4 μ μ . {\displaystyle z_{2}^{2}=z_{1}^{2}+4\mu .}

In the single parameter form, the MGF simplifies to

M ( t ) = exp ⁡ ⁡ [ μ μ ( 1 − − 1 − − 2 t ) ] . {\displaystyle M(t)=\exp[\mu (1-{\sqrt {1-2t}})].}

An inverse Gaussian distribution in double parameter form f ( x ; μ μ , λ λ ) {\displaystyle f(x;\mu ,\lambda )} can be transformed into a single parameter form f ( y ; μ μ 0 , μ μ 0 2 ) {\displaystyle f(y;\mu _{0},\mu _{0}^{2})} by appropriate scaling y = μ μ 2 x λ λ , {\displaystyle y={\frac {\mu ^{2}x}{\lambda }},} where μ μ 0 = μ μ 3 / λ λ . {\displaystyle \mu _{0}=\mu ^{3}/\lambda .}

The above paragraph can be re-written as: if Y = λ λ X / μ μ 2 {\displaystyle Y=\lambda X/\mu ^{2}} , then Y ∼ ∼ IG ⁡ ⁡ ( λ λ / μ μ , ( λ λ / μ μ ) 2 ) {\displaystyle Y\sim \operatorname {IG} (\lambda /\mu ,(\lambda /\mu )^{2})} cite-ref-folks1978-2-0[2]. This approach is better in the sense that it clearly shows dimensionless nature of the single parameter form (note that dim ⁡ ⁡ λ λ = dim ⁡ ⁡ μ μ = dim ⁡ ⁡ x {\displaystyle \dim \lambda =\dim \mu =\dim x} ). This property follows from a more general fact: if a > 0 {\displaystyle a>0} and Y = a X {\displaystyle Y=aX} , then Y ∼ ∼ IG ⁡ ⁡ ( a μ μ , a λ λ ) {\displaystyle Y\sim \operatorname {IG} (a\mu ,a\lambda )} cite-ref-tweedie1957a-3-0[3].

The standard form of inverse Gaussian distribution is

f ( x ; 1 , 1 ) = 1 2 π π x 3 exp ⁡ ⁡ ( − − ( x − − 1 ) 2 2 x ) . {\displaystyle f(x;1,1)={\frac {1}{\sqrt {2\pi x^{3}}}}\exp {\biggl (}-{\frac {(x-1)^{2}}{2x}}{\biggr )}.}

If Xi has an IG ⁡ ⁡ ( μ μ 0 w i , λ λ 0 w i 2 ) {\displaystyle \operatorname {IG} (\mu _{0}w_{i},\lambda _{0}w_{i}^{2})\,\!} distribution for i = 1, 2, ..., n and all Xi are independent, then

S = ∑ ∑ i = 1 n X i ∼ ∼ IG ⁡ ⁡ ( μ μ 0 ∑ ∑ w i , λ λ 0 ( ∑ ∑ w i ) 2 ) . {\displaystyle S=\sum _{i=1}^{n}X_{i}\sim \operatorname {IG} \left(\mu _{0}\sum w_{i},\lambda _{0}\left(\sum w_{i}\right)^{2}\right).}

Note that

Var ⁡ ⁡ ( X i ) E ⁡ ⁡ ( X i ) = μ μ 0 2 w i 2 λ λ 0 w i 2 = μ μ 0 2 λ λ 0 {\displaystyle {\frac {\operatorname {Var} (X_{i})}{\operatorname {E} (X_{i})}}={\frac {\mu _{0}^{2}w_{i}^{2}}{\lambda _{0}w_{i}^{2}}}={\frac {\mu _{0}^{2}}{\lambda _{0}}}}

is constant for all i. This is a necessary condition for the summation. Otherwise S would not be Inverse Gaussian distributed.

For any t > 0 it holds that

X ∼ ∼ IG ⁡ ⁡ ( μ μ , λ λ ) ⇒ ⇒ t X ∼ ∼ IG ⁡ ⁡ ( t μ μ , t λ λ ) . {\displaystyle X\sim \operatorname {IG} (\mu ,\lambda )\,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,tX\sim \operatorname {IG} (t\mu ,t\lambda ).}

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.

For λ λ > 0 {\displaystyle \lambda >0} fixed, it is also a single-parameter natural exponential family distributioncite-ref-seshadri1999-4-0[4] where the base distribution has density

h ( x ) = λ λ 2 π π x 3 exp ⁡ ⁡ ( − − λ λ 2 x ) 1 [ 0 , ∞ ∞ ) ( x ) . {\displaystyle h(x)={\sqrt {\frac {\lambda }{2\pi x^{3}}}}\exp \left(-{\frac {\lambda }{2x}}\right)\mathbb {1} _{[0,\infty )}(x)\,.}

Indeed, with θ θ ≤ ≤ 0 {\displaystyle \theta \leq 0} ,

p ( x ; θ θ ) = exp ⁡ ⁡ ( θ θ x ) h ( x ) ∫ ∫ exp ⁡ ⁡ ( θ θ y ) h ( y ) d y {\displaystyle p(x;\theta )={\frac {\exp(\theta x)h(x)}{\int \exp(\theta y)h(y)dy}}}

is a density over the reals. Evaluating the integral, we get

p ( x ; θ θ ) = λ λ 2 π π x 3 exp ⁡ ⁡ ( − − λ λ 2 x + θ θ x − − − − 2 λ λ θ θ ) 1 [ 0 , ∞ ∞ ) ( x ) . {\displaystyle p(x;\theta )={\sqrt {\frac {\lambda }{2\pi x^{3}}}}\exp \left(-{\frac {\lambda }{2x}}+\theta x-{\sqrt {-2\lambda \theta }}\right)\mathbb {1} _{[0,\infty )}(x)\,.}

Substituting θ θ = − − λ λ / ( 2 μ μ 2 ) {\displaystyle \theta =-\lambda /(2\mu ^{2})} makes the above expression equal to f ( x ; μ μ , λ λ ) {\displaystyle f(x;\mu ,\lambda )} .

Let the stochastic process Xt be given by

X 0 = 0 {\displaystyle X_{0}=0\quad }

X t = ν ν t + σ σ W t {\displaystyle X_{t}=\nu t+\sigma W_{t}\quad \quad \quad \quad }

where Wt is a standard Brownian motion. That is, Xt is a Brownian motion with drift ν ν > 0 {\displaystyle \nu >0} .

Then the first passage time for a fixed level α α > 0 {\displaystyle \alpha >0} by Xt is distributed according to an inverse-Gaussian:

T α α = inf { t > 0 ∣ ∣ X t = α α } ∼ ∼ IG ⁡ ⁡ ( α α ν ν , ( α α σ σ ) 2 ) = α α σ σ 2 π π x 3 exp ⁡ ⁡ ( − − ( α α − − ν ν x ) 2 2 σ σ 2 x ) {\displaystyle T_{\alpha }=\inf\{t>0\mid X_{t}=\alpha \}\sim \operatorname {IG} \left({\frac {\alpha }{\nu }},\left({\frac {\alpha }{\sigma }}\right)^{2}\right)={\frac {\alpha }{\sigma {\sqrt {2\pi x^{3}}}}}\exp {\biggl (}-{\frac {(\alpha -\nu x)^{2}}{2\sigma ^{2}x}}{\biggr )}}

i.e

P ( T α α ∈ ∈ ( T , T + d T ) ) = α α σ σ 2 π π T 3 exp ⁡ ⁡ ( − − ( α α − − ν ν T ) 2 2 σ σ 2 T ) d T {\displaystyle P(T_{\alpha }\in (T,T+dT))={\frac {\alpha }{\sigma {\sqrt {2\pi T^{3}}}}}\exp {\biggl (}-{\frac {(\alpha -\nu T)^{2}}{2\sigma ^{2}T}}{\biggr )}dT}

(cf. Schrödingercite-ref-schr-dinger1915-5-0[5] equation 19, Smoluchowskicite-ref-smoluchowski1915-6-0[6], equation 8, and Folkscite-ref-folks1978-2-1[2], equation 1).

A common special case of the above arises when the Brownian motion has no drift. In that case, parameter μ tends to infinity, and the first passage time for fixed level α has probability density function

f ( x ; 0 , ( α α σ σ ) 2 ) = α α σ σ 2 π π x 3 exp ⁡ ⁡ ( − − α α 2 2 σ σ 2 x ) {\displaystyle f\left(x;0,\left({\frac {\alpha }{\sigma }}\right)^{2}\right)={\frac {\alpha }{\sigma {\sqrt {2\pi x^{3}}}}}\exp \left(-{\frac {\alpha ^{2}}{2\sigma ^{2}x}}\right)}

(see also Bacheliercite-ref-bachelier1900a-7-0[7]cite-ref-bachelier1900b-8-0[8]). This is a Lévy distribution with parameters c = ( α α σ σ ) 2 {\displaystyle c=\left({\frac {\alpha }{\sigma }}\right)^{2}} and μ μ = 0 {\displaystyle \mu =0} .

The model where

X i ∼ ∼ IG ⁡ ⁡ ( μ μ , λ λ w i ) , i = 1 , 2 , … … , n {\displaystyle X_{i}\sim \operatorname {IG} (\mu ,\lambda w_{i}),\,\,\,\,\,\,i=1,2,\ldots ,n}

with all wi known, (μ, λ) unknown and all Xi independent has the following likelihood function

L ( μ μ , λ λ ) = ( λ λ 2 π π ) n 2 ( ∏ ∏ i = 1 n w i X i 3 ) 1 2 exp ⁡ ⁡ ( λ λ μ μ ∑ ∑ i = 1 n w i − − λ λ 2 μ μ 2 ∑ ∑ i = 1 n w i X i − − λ λ 2 ∑ ∑ i = 1 n w i 1 X i ) . {\displaystyle L(\mu ,\lambda )=\left({\frac {\lambda }{2\pi }}\right)^{\frac {n}{2}}\left(\prod _{i=1}^{n}{\frac {w_{i}}{X_{i}^{3}}}\right)^{\frac {1}{2}}\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}{\frac {1}{X_{i}}}\right).}

Solving the likelihood equation yields the following maximum likelihood estimates

μ μ ^ ^ = ∑ ∑ i = 1 n w i X i ∑ ∑ i = 1 n w i , 1 λ λ ^ ^ = 1 n ∑ ∑ i = 1 n w i ( 1 X i − − 1 μ μ ^ ^ ) . {\displaystyle {\widehat {\mu }}={\frac {\sum _{i=1}^{n}w_{i}X_{i}}{\sum _{i=1}^{n}w_{i}}},\,\,\,\,\,\,\,\,{\frac {1}{\widehat {\lambda }}}={\frac {1}{n}}\sum _{i=1}^{n}w_{i}\left({\frac {1}{X_{i}}}-{\frac {1}{\widehat {\mu }}}\right).}

μ μ ^ ^ {\displaystyle {\widehat {\mu }}} and λ λ ^ ^ {\displaystyle {\widehat {\lambda }}} are independent and

μ μ ^ ^ ∼ ∼ IG ⁡ ⁡ ( μ μ , λ λ ∑ ∑ i = 1 n w i ) , n λ λ ^ ^ ∼ ∼ 1 λ λ χ χ n − − 1 2 . {\displaystyle {\widehat {\mu }}\sim \operatorname {IG} \left(\mu ,\lambda \sum _{i=1}^{n}w_{i}\right),\qquad {\frac {n}{\widehat {\lambda }}}\sim {\frac {1}{\lambda }}\chi _{n-1}^{2}.}

The following algorithm may be used.cite-ref-michael1976-9-0[9]

Generate a random variate from a normal distribution with mean 0 and standard deviation equal 1 ν ν ∼ ∼ N ( 0 , 1 ) . {\displaystyle \displaystyle \nu \sim N(0,1).} Square the value y = ν ν 2 {\displaystyle \displaystyle y=\nu ^{2}} and use the relation x = μ μ + μ μ 2 y 2 λ λ − − μ μ 2 λ λ 4 μ μ λ λ y + μ μ 2 y 2 . {\displaystyle x=\mu +{\frac {\mu ^{2}y}{2\lambda }}-{\frac {\mu }{2\lambda }}{\sqrt {4\mu \lambda y+\mu ^{2}y^{2}}}.} Generate another random variate, this time sampled from a uniform distribution between 0 and 1 z ∼ ∼ U ( 0 , 1 ) . {\displaystyle \displaystyle z\sim U(0,1).} If z ≤ ≤ μ μ μ μ + x {\displaystyle z\leq {\frac {\mu }{\mu +x}}} then return x {\displaystyle \displaystyle x} else return μ μ 2 x . {\displaystyle {\frac {\mu ^{2}}{x}}.}

Sample code in Java:

And to plot Wald distribution in Python using matplotlib and NumPy:

• If X ∼ ∼ IG ⁡ ⁡ ( μ μ , λ λ ) {\displaystyle X\sim \operatorname {IG} (\mu ,\lambda )} , then k X ∼ ∼ IG ⁡ ⁡ ( k μ μ , k λ λ ) {\displaystyle kX\sim \operatorname {IG} (k\mu ,k\lambda )} for any number k > 0. {\displaystyle k>0.} cite-ref-chhikara1989-1-1[1]

• If X i ∼ ∼ IG ⁡ ⁡ ( μ μ , λ λ ) {\displaystyle X_{i}\sim \operatorname {IG} (\mu ,\lambda )\,} then ∑ ∑ i = 1 n X i ∼ ∼ IG ⁡ ⁡ ( n μ μ , n 2 λ λ ) {\displaystyle \sum _{i=1}^{n}X_{i}\sim \operatorname {IG} (n\mu ,n^{2}\lambda )\,}

• If X i ∼ ∼ IG ⁡ ⁡ ( μ μ , λ λ ) {\displaystyle X_{i}\sim \operatorname {IG} (\mu ,\lambda )\,} for i = 1 , … … , n {\displaystyle i=1,\ldots ,n\,} then X ¯ ¯ ∼ ∼ IG ⁡ ⁡ ( μ μ , n λ λ ) {\displaystyle {\bar {X}}\sim \operatorname {IG} (\mu ,n\lambda )\,}

• If X i ∼ ∼ IG ⁡ ⁡ ( μ μ i , 2 μ μ i 2 ) {\displaystyle X_{i}\sim \operatorname {IG} (\mu _{i},2\mu _{i}^{2})\,} then ∑ ∑ i = 1 n X i ∼ ∼ IG ⁡ ⁡ ( ∑ ∑ i = 1 n μ μ i , 2 ( ∑ ∑ i = 1 n μ μ i ) 2 ) {\displaystyle \sum _{i=1}^{n}X_{i}\sim \operatorname {IG} \left(\sum _{i=1}^{n}\mu _{i},2\left(\sum _{i=1}^{n}\mu _{i}\right)^{2}\right)\,}

• If X ∼ ∼ IG ⁡ ⁡ ( μ μ , λ λ ) {\displaystyle X\sim \operatorname {IG} (\mu ,\lambda )} , then λ λ ( X − − μ μ ) 2 / μ μ 2 X ∼ ∼ χ χ 2 ( 1 ) {\displaystyle \lambda (X-\mu )^{2}/\mu ^{2}X\sim \chi ^{2}(1)} .cite-ref-schuster1968-10-0[10]

The convolution of an inverse Gaussian distribution (a Wald distribution) and an exponential (an ex-Wald distribution) is used as a model for response times in psychology,cite-ref-schwarz2001-11-0[11] with visual search as one example.cite-ref-palmer2010-12-0[12]

This distribution appears to have been first derived in 1900 by Louis Bacheliercite-ref-bachelier1900a-7-1[7]cite-ref-bachelier1900b-8-1[8] as the time a stock reaches a certain price for the first time. In 1915 it was used independently by Erwin Schrödingercite-ref-schr-dinger1915-5-1[5] and Marian v. Smoluchowskicite-ref-smoluchowski1915-6-1[6] as the time to first passage of a Brownian motion. In the field of reproduction modeling it is known as the Hadwiger function, after Hugo Hadwiger who described it in 1940.cite-ref-harwiger1940-13-0[13] Abraham Wald re-derived this distribution in 1944cite-ref-wald1944-14-0[14] as the limiting form of a sample in a sequential probability ratio test. The name inverse Gaussian was proposed by Maurice Tweedie in 1945.cite-ref-tweedie1945-15-0[15] Tweedie investigated this distribution in 1956cite-ref-tweedie1956-16-0[16] and 1957cite-ref-tweedie1957a-3-1[3]cite-ref-tweedie1957b-17-0[17] and established some of its statistical properties. The distribution was extensively reviewed by Folks and Chhikara in 1978.cite-ref-folks1978-2-2[2]

Assuming that the time intervals between occurrences of a random phenomenon follow an inverse Gaussian distribution, the probability distribution for the number of occurrences of this event within a specified time window is referred to as rated inverse Gaussian.cite-ref-18[18] While, first and second moment of this distribution are calculated, the derivation of the moment generating function remains an open problem.

Despite the simple formula for the probability density function, numerical probability calculations for the inverse Gaussian distribution nevertheless require special care to achieve full machine accuracy in floating point arithmetic for all parameter values.cite-ref-giner2016-19-0[19] Functions for the inverse Gaussian distribution are provided for the R programming language by several packages including rmutil,cite-ref-20[20]cite-ref-21[21] SuppDists,cite-ref-22[22] STAR,cite-ref-23[23] invGauss,cite-ref-24[24] LaplacesDemon,cite-ref-25[25] and statmod.cite-ref-26[26]

• Generalized inverse Gaussian distribution

• Tweedie distributions—The inverse Gaussian distribution is a member of the family of Tweedie exponential dispersion models

• Stopping time

cite-note-chhikara1989-11. ↑ citerefchhikarafolks1989Chhikara, Raj S.; Folks, J. Leroy (1989), The Inverse Gaussian Distribution: Theory, Methodology and Applications, New York, NY, USA: Marcel Dekker, Inc, ISBN 0-8247-7997-5

cite-note-folks1978-22. ↑ citereffolkschhikara1978Folks, J. Leroy; Chhikara, Raj S. (1978), "The Inverse Gaussian Distribution and Its Statistical Application—A Review", Journal of the Royal Statistical Society, Series B (Methodological), 40 (3): 263–275, doi:10.1111/j.2517-6161.1978.tb01039.x, JSTOR 2984691, S2CID 125337421

cite-note-tweedie1957a-33. ↑ citereftweedie1957Tweedie, M. C. K. (1957). "Statistical Properties of Inverse Gaussian Distributions I". Annals of Mathematical Statistics. 28 (2): 362–377. doi:10.1214/aoms/1177706964. JSTOR 2237158.

cite-note-seshadri1999-44. ↑ citerefseshadri1999Seshadri, V. (1999), The Inverse Gaussian Distribution, Springer-Verlag, ISBN 978-0-387-98618-0

cite-note-schr-dinger1915-55. ↑ citerefschr-dinger1915Schrödinger, Erwin (1915), "Zur Theorie der Fall- und Steigversuche an Teilchen mit Brownscher Bewegung" [On the Theory of Fall- and Rise Experiments on Particles with Brownian Motion], Physikalische Zeitschrift (in German), 16 (16): 289–295

cite-note-smoluchowski1915-66. ↑ citerefsmoluchowski1915Smoluchowski, Marian (1915), "Notiz über die Berechnung der Brownschen Molekularbewegung bei der Ehrenhaft-Millikanschen Versuchsanordnung" [Note on the Calculation of Brownian Molecular Motion in the Ehrenhaft-Millikan Experimental Set-up], Physikalische Zeitschrift (in German), 16 (17/18): 318–321

cite-note-bachelier1900a-77. ↑ citerefbachelier1900Bachelier, Louis (1900), "Théorie de la spéculation" [The Theory of Speculation] (PDF), Ann. Sci. Éc. Norm. Supér. (in French), Serie 3, 17: 21–89, doi:10.24033/asens.476

cite-note-bachelier1900b-88. ↑ citerefbachelier1900Bachelier, Louis (1900), "The Theory of Speculation", Ann. Sci. Éc. Norm. Supér., Serie 3, 17: 21–89 (Engl. translation by David R. May, 2011), doi:10.24033/asens.476

cite-note-michael1976-99. ↑ citerefmichaelschucanyhaas1976Michael, John R.; Schucany, William R.; Haas, Roy W. (1976), "Generating Random Variates Using Transformations with Multiple Roots", The American Statistician, 30 (2): 88–90, doi:10.1080/00031305.1976.10479147, JSTOR 2683801

cite-note-schuster1968-1010. ↑ citerefshuster1968Shuster, J. (1968). "On the inverse Gaussian distribution function". Journal of the American Statistical Association. 63 (4): 1514–1516. doi:10.1080/01621459.1968.10480942.

cite-note-schwarz2001-1111. ↑ citerefschwarz2001Schwarz, Wolfgang (2001), "The ex-Wald distribution as a descriptive model of response times", Behavior Research Methods, Instruments, and Computers, 33 (4): 457–469, doi:10.3758/bf03195403, PMID 11816448

cite-note-palmer2010-1212. ↑ citerefpalmerhorowitztorralbawolfe2011Palmer, E. M.; Horowitz, T. S.; Torralba, A.; Wolfe, J. M. (2011). "What are the shapes of response time distributions in visual search?". Journal of Experimental Psychology: Human Perception and Performance. 37 (1): 58–71. doi:10.1037/a0020747. PMC 3062635. PMID 21090905.

cite-note-harwiger1940-1313. ↑ citerefhadwiger1940Hadwiger, H. (1940). "Eine analytische Reproduktionsfunktion für biologische Gesamtheiten". Skandinavisk Aktuarietidskrijt. 7 (3–4): 101–113. doi:10.1080/03461238.1940.10404802.

cite-note-wald1944-1414. ↑ citerefwald1944Wald, Abraham (1944), "On Cumulative Sums of Random Variables", Annals of Mathematical Statistics, 15 (3): 283–296, doi:10.1214/aoms/1177731235, JSTOR 2236250

cite-note-tweedie1945-1515. ↑ citereftweedie1945Tweedie, M. C. K. (1945). "Inverse Statistical Variates". Nature. 155 (3937): 453. Bibcode:1945Natur.155..453T. doi:10.1038/155453a0. S2CID 4113244.

cite-note-tweedie1956-1616. ↑ citereftweedie1956Tweedie, M. C. K. (1956). "Some Statistical Properties of Inverse Gaussian Distributions". Virginia Journal of Science. New Series. 7 (3): 160–165.

cite-note-tweedie1957b-1717. ↑ citereftweedie1957Tweedie, M. C. K. (1957). "Statistical Properties of Inverse Gaussian Distributions II". Annals of Mathematical Statistics. 28 (3): 696–705. doi:10.1214/aoms/1177706881. JSTOR 2237229.

cite-note-1818. ↑ Capacity per unit cost-achieving input distribution of rated-inverse gaussian biological neuron M Nasiraee, HM Kordy, J Kazemitabar IEEE Transactions on Communications 70 (6), 3788-3803

cite-note-giner2016-1919. ↑ citerefginersmyth2016Giner, Göknur; Smyth, Gordon (August 2016). "statmod: Probability Calculations for the Inverse Gaussian Distribution". The R Journal. 8 (1): 339–351. arXiv:1603.06687. doi:10.32614/RJ-2016-024.

cite-note-2020. ↑ citereflindsey2013Lindsey, James (2013-09-09). "rmutil: Utilities for Nonlinear Regression and Repeated Measurements Models".

cite-note-2121. ↑ citerefswihartlindsey2019Swihart, Bruce; Lindsey, James (2019-03-04). "rmutil: Utilities for Nonlinear Regression and Repeated Measurements Models".

cite-note-2222. ↑ citerefwheeler2016Wheeler, Robert (2016-09-23). "SuppDists: Supplementary Distributions".

cite-note-2323. ↑ citerefpouzat2015Pouzat, Christophe (2015-02-19). "STAR: Spike Train Analysis with R".

cite-note-2424. ↑ citerefgjessing2014Gjessing, Hakon K. (2014-03-29). "Threshold regression that fits the (randomized drift) inverse Gaussian distribution to survival data".

cite-note-2525. ↑ citerefhallhallstatisticatbrown2014Hall, Byron; Hall, Martina; Statisticat, LLC; Brown, Eric; Hermanson, Richard; Charpentier, Emmanuel; Heck, Daniel; Laurent, Stephane; Gronau, Quentin F.; Singmann, Henrik (2014-03-29). "LaplacesDemon: Complete Environment for Bayesian Inference".

cite-note-2626. ↑ citerefginersmyth2017Giner, Göknur; Smyth, Gordon (2017-06-18). "statmod: Statistical Modeling".

• citerefh-ylandrausand1994Høyland, Arnljot; Rausand, Marvin (1994). System Reliability Theory. New York: Wiley. ISBN 978-0-471-59397-3.

• citerefseshadri1993Seshadri, V. (1993). The Inverse Gaussian Distribution. Oxford University Press. ISBN 978-0-19-852243-0.

• Inverse Gaussian Distribution in Wolfram website.