laplace.rst

Laplace Distribution

Table of contents

local

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Density Function

The density function of the Laplace distribution:

$$f(x; \mu, \sigma) = \frac{1}{2 \sigma} \exp \left( - \frac{|x-\mu|}{\sigma} \right)$$

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 Laplace distribution:

$$F(x; \mu, \sigma) = \int_{-\infty}^x f(z; \mu, \sigma) dz = \frac{1}{2} + \frac{1}{2} \times \text{sign} ( x - \mu ) \times \left( 1 - \exp \left( - \frac{|x - \mu|}{\sigma} \right) \right)$$

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 Laplace distribution:

q(p; μ, σ) = μ − σ × sign(p − 0.5) × ln (1 − 2|p − 0.5|)

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

Random sampling for the Laplace distribution is achieved via the inverse probability integral transform.

Scalar Output

1. Random number engines
2. Seed values

Vector/Matrix Output

1. Random number engines
2. Seed values