Laplace Distribution

Table of contents

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 list input, are listed below.

Scalar Input

pystats.dlaplace(x: float, mu: float = 0.0, sigma: float = 1.0, log: bool = False) → float

Density function of the Laplace distribution.

Example

>>> pystats.dlaplace(0.7, 1.0, 2.0)
0.21517699410626442
Parameters

  • x (float) – A real-valued input.

  • mu (float) – The location parameter, a real-valued input.

  • sigma (float) – The scale parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

The density function evaluated at x.

List Input

pystats.dlaplace(x: List[float], mu: float = 0.0, sigma: float = 1.0, log: bool = False) → List[float]

Density function of the Laplace distribution.

Example

>>> pystats.dlaplace([0.0, 1.0, 2.0], 1.0, 2.0)
[0.15163266492815836, 0.25, 0.15163266492815836]
Parameters

  • x (List[float]) – A standard list input.

  • mu (float) – The location parameter, a real-valued input.

  • sigma (float) – The scale parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

A list of density values corresponding to the elements of x.

Cumulative Distribution Function

The cumulative distribution function (CDF) 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 list input, are listed below.

Scalar Input

pystats.plaplace(p: float, mu: float = 0.0, sigma: float = 1.0, log: bool = False) → float

Distribution function of the Laplace distribution.

Example

>>> pystats.plaplace(0.7, 1.0, 2.0)
0.4303539882125289
Parameters

  • p (float) – A real-valued input.

  • mu (float) – The location parameter, a real-valued input.

  • sigma (float) – The scale parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

The cumulative distribution function evaluated at p.

List Input

pystats.plaplace(p: List[float], mu: float = 0.0, sigma: float = 1.0, log: bool = False) → List[float]

Distribution function of the Laplace distribution.

Example

>>> pystats.plaplace([0.0, 1.0, 2.0], 1.0, 2.0)
[0.3032653298563167, 0.5, 0.6967346701436833]
Parameters

  • p (List[float]) – A standard list input.

  • mu (float) – The location parameter, a real-valued input.

  • sigma (float) – The scale parameter, a real-valued input.

  • log (bool) – Return the log-density or the true form.

Returns

A list of CDF values corresponding to the elements of p.

Quantile Function

The quantile function of the Laplace distribution:

\[q(p; \mu, \sigma) = \mu - \sigma \times \text{sign}(p - 0.5) \times \ln(1 - 2 | p - 0.5 |)\]

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.qlaplace(q: float, mu: float = 0.0, sigma: float = 1.0) → float

Quantile function of the Laplace distribution.

Example

>>> pystats.qlaplace(0.5, 1.0, 2.0)
1.0
Parameters

  • q (float) – A real-valued input.

  • mu (float) – The location parameter, a real-valued input.

  • sigma (float) – The scale parameter, a real-valued input.

Returns

The quantile function evaluated at q.

List Input

pystats.qlaplace(q: List[float], mu: float = 0.0, sigma: float = 1.0) → List[float]

Quantile function of the Laplace distribution.

Example

>>> pystats.qlaplace([0.1, 0.6, 0.9], 1.0, 2.0)
[-2.218875824868202, 1.4462871026284194, 4.218875824868202]
Parameters

  • q (List[float]) – A standard list input.

  • mu (float) – The location parameter, a real-valued input.

  • sigma (float) – The scale parameter, a real-valued input.

Returns

A list of quantiles values corresponding to the elements of q.

Random Sampling

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

Scalar Output

pystats.rlaplace(mu: float = 0.0, sigma: float = 1.0) → float

Random sampling function for the Laplace distribution.

Example

>>> pystats.rlaplace(1.0, 2.0)
-0.06931476521956581
Parameters

  • mu (float) – The location parameter, a real-valued input.

  • sigma (float) – The scale parameter, a real-valued input.

Returns

A pseudo-random draw from the Laplace distribution.

List Output

pystats.rlaplace(n: int, mu: float = 0.0, sigma: float = 1.0) → List[float]

Random sampling function for the Laplace distribution.

Example

>>> pystats.rlaplace(3, 1.0, 2.0)
[0.7785860236771809, 0.14724988410675632, 3.0446760367044163]
Parameters

  • n (int) – The number of output values.

  • mu (float) – The location parameter, a real-valued input.

  • sigma (float) – The scale parameter, a real-valued input.

Returns

A list of pseudo-random draws from the Laplace distribution.