inverse-gaussian.rst

Inverse-Gaussian Distribution

Table of contents

local

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Density Function

The density function of the inverse Gaussian distribution:

$$f(x; \mu, \lambda) = \sqrt{ \dfrac{\lambda}{2 \pi x^3} } \exp \left[ - \dfrac{\lambda(x - \mu)^2}{2 \mu^2 x} \right] \times \mathbf{1}[ x \geq 0 ]$$

where μ is the mean parameter and λ is the shape parameter.

Methods for scalar input, as well as for vector/matrix input, are listed below.

Scalar Input

Vector/Matrix Input

STL Containers

Armadillo

Blaze

Eigen

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Cumulative Distribution Function

The cumulative distribution function of the inverse Gaussian distribution:

$$F(x; \mu, \lambda) = \Phi \left( \sqrt{\dfrac{\lambda}{x}} \left( \frac{x}{\mu} - 1 \right) \right) + \exp \left( \dfrac{2 \lambda}{\mu} \right) \Phi \left( - \sqrt{\dfrac{\lambda}{x}} \left( \frac{x}{\mu} + 1 \right) \right)$$

where Φ( ⋅ ) denotes the standard Normal CDF.

Methods for scalar input, as well as for vector/matrix input, are listed below.

Scalar Input

Vector/Matrix Input

STL Containers

Armadillo

Blaze

Eigen

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Quantile Function

The quantile function of the inverse Gaussian distribution:

q(p; μ, λ) = inf {x:p≤F(x;μ,λ)}

where F( ⋅ ) denotes the CDF of the inverse Gaussian distribution.

Methods for scalar input, as well as for vector/matrix input, are listed below.

Scalar Input

Vector/Matrix Input

STL Containers

Armadillo

Blaze

Eigen

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Random Sampling

Scalar Output

1. Random number engines
2. Seed values

Vector/Matrix Output

1. Random number engines
2. Seed values