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