# Distributions

Functions for probability distributions and draws.

## biogeme.distributions module

Implementation of the pdf and CDF of common distributions

author:

Michel Bierlaire

date:

Thu Apr 23 12:01:49 2015

biogeme.distributions.logisticcdf(x, mu=0.0, s=1.0)[source]

Logistic CDF

Cumulative distribution function of a logistic distribution

$f(x;\mu, \sigma) = \frac{1} {1+\exp\left(-\frac{x-\mu}{\sigma} \right)}$
Parameters:
• x (float or biogeme.expression) – value at which the CDF is evaluated.

• mu (float or biogeme.expression) – location parameter $$\mu$$ of the logistic distribution. Default: 0.

• s (float or biogeme.expression) – scale parameter $$\sigma$$ of the logistic distribution. Default: 1.

Note:

It is assumed that $$\sigma > 0$$, but it is not verified by the code.

Returns:

value of the logistic CDF.

Return type:

float or biogeme.expression

biogeme.distributions.lognormalpdf(x, mu=0.0, s=1.0)[source]

Log normal pdf

Probability density function of a log normal distribution

$f(x;\mu, \sigma) = \frac{1}{x\sigma \sqrt{2\pi}} \exp{-\frac{(\ln x-\mu)^2}{2\sigma^2}}$
Parameters:
• x (float or biogeme.expression) – value at which the pdf is evaluated.

• mu (float or biogeme.expression) – location parameter $$\mu$$ of the lognormal distribution. Default: 0.

• s (float or biogeme.expression) – scale parameter $$\sigma$$ of the lognormal distribution. Default: 1.

Note:

It is assumed that $$\sigma > 0$$, but it is not verified by the code.

Returns:

value of the lognormal pdf.

Return type:

float or biogeme.expression

biogeme.distributions.normalpdf(x, mu=0.0, s=1.0)[source]

Normal pdf

Probability density function of a normal distribution

$f(x;\mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp{-\frac{(x-\mu)^2}{2\sigma^2}}$
Parameters:
• x (float or biogeme.expression) – value at which the pdf is evaluated.

• mu (float or biogeme.expression) – location parameter $$\mu$$ of the Normal distribution. Default: 0.

• s (float or biogeme.expression) – scale parameter $$\sigma$$ of the Normal distribution. Default: 1.

Note:

It is assumed that $$\sigma > 0$$, but it is not verified by the code.

Returns:

value of the Normal pdf.

Return type:

float or biogeme.expression

biogeme.distributions.triangularpdf(x, a=-1.0, b=1.0, c=0.0)[source]

Triangular pdf

Probability density function of a triangular distribution

$\begin{split}f(x;a, b, c) = \left\{ \begin{array}{ll} 0 & \text{if } x < a \\\frac{2(x-a)}{(b-a)(c-a)} & \text{if } a \leq x < c \\\frac{2(b-x)}{(b-a)(b-c)} & \text{if } c \leq x < b \\0 & \text{if } x \geq b. \end{array} \right.\end{split}$
Parameters:
• x (float or biogeme.expression) – argument of the pdf

• a (float or biogeme.expression) – lower bound $$a$$ of the distribution. Default: -1.

• b (float or biogeme.expression) – upper bound $$b$$ of the distribution. Default: 1.

• c (float or biogeme.expression) – mode $$c$$ of the distribution. Default: 0.

Note:

It is assumed that $$a < c < b$$, but it is not verified by the code.

Returns:

value of the triangular pdf.

Return type:

float or biogeme.expression

biogeme.distributions.uniformpdf(x, a=-1, b=1.0)[source]

Uniform pdf

Probability density function of a uniform distribution.

$\begin{split}f(x;a, b) = \left\{ \begin{array}{ll} \frac{1}{b-a} & \text{for } x \in [a, b] \\ 0 & \text{otherwise}\end{array} \right.\end{split}$
Parameters:
• x (float or biogeme.expression) – argument of the pdf

• a (float or biogeme.expression) – lower bound $$a$$ of the distribution. Default: -1.

• b (float or biogeme.expression) – upper bound $$b$$ of the distribution. Default: 1.

Note:

It is assumed that $$a < b$$, but it is not verified by the code.

Returns:

value of the uniform pdf.

Return type:

float or biogeme.expression