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Encyclopedia > Inverse Gaussian distribution
Inverse Gaussian
Probability density function
Cumulative distribution function
Parameters λ > 0
μ > 0
Support  x in (0,infty)
Probability density function (pdf)  left[frac{lambda}{2 pi x^3}right]^{1/2} exp{frac{-lambda (x-mu)^2}{2 mu^2 x}}
Cumulative distribution function (cdf)  Phileft(sqrt{frac{lambda}{x}} left(frac{x}{mu}-1 right)right) +expleft(frac{2 lambda}{mu}right) Phileft(-sqrt{frac{lambda}{x}}left(frac{x}{mu}+1 right)right)

where  Phi left(right) is the normal (Gaussian) distribution c.d.f. Image File history File links Size of this preview: 776 × 600 pixelsFull resolution (1100 × 850 pixel, file size: 11 KB, MIME type: image/png) GNU R source code png(PDF_invGauss. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...

Mean μ
Median
Mode muleft[left(1+frac{9 mu^2}{4 lambda^2}right)^frac{1}{2}-frac{3 mu}{2 lambda}right]
Variance frac{mu^3}{lambda}
Skewness 3left(frac{mu}{lambda}right)^{1/2}
Excess kurtosis 3 +frac{15 mu}{lambda}
Entropy
Moment-generating function (mgf) e^{left(frac{lambda}{mu}right)left[1-sqrt{1-frac{2mu^2t}{lambda}}right]}
Characteristic function e^{left(frac{lambda}{mu}right)left[1-sqrt{1-frac{2mu^2mathrm{i}t}{lambda}}right]}

The probability density function of the inverse Gaussian distribution is given by In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

 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.

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 a Brownian Motion with positive drift takes to reach a fixed positive level. The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. ...


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 A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...

X sim IG(mu, lambda).,!

Contents

Properties

Summation

If Xi has a IG(μ0wi, λ0wi2) distribution for i = 1, 2, ..., n, then

provided all Xi are independent.


Note that

 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}

is constant for all i. This is a nessesary condition for the summation. Otherwise S would not be inverse gaussian. This article discusses only the formal meanings of necessary and sufficient causal meanings see causation. ...


Scaling

For any t > 0 it holds that

 X sim IG(mu,lambda) ,,,,,, Rightarrow ,,,,,, tX sim IG(tmu,tlambda)

Exponential family

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. In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ...


Inverse Gaussian distribution and Brownian motion

The relationship between the inverse Gaussian distribution and Brownian motion is as follows: The stochastic process Xt given by In the mathematics of probability, a stochastic process is a random function. ...

(where Wt is a standard Brownian motion) is a Brownian motion with drift ν. The first passage time for a fixed level α > 0 by Xt is

If ν > 0 then

Maximum likelihood

The model where

 X_i sim 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

Solving the likelihood equation yields the following maximum likelihood estimates

 hat{mu}= frac{sum_{i=1}^n w_i X_i}{sum_{i=1}^n w_i}, ,,,,,,,, frac1hat{lambda}= frac1n sum_{i=1}^n w_i left( frac1{X_i}-frac1{hat{mu}} right)

hat{mu} and hat{lambda} are independent and

References

  • 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
  • The Inverse Gaussian Distribution by D.N. Seshadri, Oxford Univ Press

See also

In probability theory, the Generalized inverse Gaussian distribution (GIG) is a probability distribution with probability density function It is used extensively in geostatistics, statistical linguistics, finance, etc. ...

External link

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