Inverse Gaussian Distribution

The inverse Gaussian distribution, also known as the Wald distribution, is a two-parameter family of continuous probability distributions. It is related to the normal distribution but is not the distribution of the reciprocal of a normal variable.

The inverse Gaussian distribution has a location parameter $\mu$ and a scale parameter $\lambda$. The probability density function (PDF) is given by:

The cumulative distribution function (CDF) is:

where $\mathrm{\Phi }(x)$ is the standard normal CDF.

The inverse Gaussian distribution is defined for all positive real numbers (x > 0), with both its location parameter μ and scale parameter λ required to be strictly positive as well.

• Survival analysis uses this distribution to model time-to-event data with positive skewness.

• Financial modeling employs the distribution for analyzing asset prices and interest rates.

• Reliability engineering uses it to model failure times in degradation processes.

• Physics applications include modeling Brownian motion with drift.

Notable properties of the inverse Gaussian distribution include:

• The distribution is unimodal and positively skewed.

• The distribution exhibits infinite divisibility.

• The inverse Gaussian distribution approaches a normal distribution as $\lambda \to \mathrm{\infty }$.

• The distribution is a member of the exponential family.

• It is related to the Lévy distribution when $\mu \to \mathrm{\infty }$.

The inverse Gaussian distribution is implemented by the InverseGaussianDistribution class. It has just one constructor with two arguments. The first argument is the shape parameter. The second argument is the scale parameter.

The following constructs an inverse Gaussian distribution of order 4.2 and scale parameter 1:

var invGaussian = new InverseGaussianDistribution(4.2, 1.0);

Dim invGaussian = New InverseGaussianDistribution(4.2, 1.0)

No code example is currently available or this language may not be supported.

let invGaussian = InverseGaussianDistribution(4.2, 1.0)

The InverseGaussianDistribution class has two specific properties, LocationParameter and ScaleParameter, which return the location and scale parameters of the distribution.

For more information on the inverse Gaussian distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 15. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 25. Wiley Series in Probability and Statistics.

• Chhikara, R. S., & Folks, J. L. (1989). The Inverse Gaussian Distribution: Theory, Methodology, and Applications. CRC Press.

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