Modelica.Math.Distributions.Normal

Library of normal distribution functions

Information

This package provides

• probability density function (= derivative of cumulative distribution function),
• cumulative distribution function, and
• quantile (= inverse cumulative distribution function).

of the normal distribution. Examples:

For more details of this distribution see Wikipedia.

Extends from Modelica.Icons.Package (Icon for standard packages).

Package Content

Name	Description
density	Density of normal distribution
cumulative	Cumulative distribution function of normal distribution
quantile	Quantile of normal distribution

Modelica.Math.Distributions.Normal.density

Density of normal distribution

Information

Syntax

Normal.density(u, mu=0, sigma=1);

Description

This function computes the probability density function according to a normal distribution with mean value mu and standard deviation sigma (variance = sigma²). Plot of the function:

For more details, see Wikipedia.

Example

  density(0.5)     // = 0.3520653267642995
  density(3,1,0.5) // = 0.00026766045152977074

See also

Normal.cumulative, Normal.quantile.

Extends from Modelica.Math.Distributions.Interfaces.partialDensity (Common interface of probability density functions).

Inputs

Outputs

Modelica.Math.Distributions.Normal.cumulative

Cumulative distribution function of normal distribution

Information

Syntax

Normal.cumulative(u, mu=0, sigma=1);

Description

This function computes the cumulative distribution function according to a normal distribution with mean value mu and standard deviation sigma (variance = sigma²). The returned value y is in the range:

0 ≤ y ≤ 1

Plot of the function:

For more details, see Wikipedia.

Example

  cumulative(0.5)      // = 0.6914624612740131
  cumulative(0,1,0.5)  // = 0.15865525393145707

See also

Normal.density, Normal.quantile.

Extends from Modelica.Math.Distributions.Interfaces.partialCumulative (Common interface of cumulative distribution functions).

Inputs

Outputs

Modelica.Math.Distributions.Normal.quantile

Quantile of normal distribution

Information

Syntax

Normal.quantile(u, y_min=0, y_max=1);

Description

This function computes the inverse cumulative distribution function (= quantile) according to a normal distribution with mean value mu and standard deviation sigma (variance = sigma²). Input argument u must be in the range:

0 < u < 1

If the input argument u is a uniformly distributed random number, then 99.7 % of the returned random numbers are in the range:

mu-3*sigma ≤ y ≤ mu+3*sigma

Plot of the function:

For more details, see Wikipedia.

Example

  quantile(0.001)     // = -3.090232306167813;
  quantile(0.5,1,0.5) // = 1

See also

Normal.density, Normal.cumulative.

Extends from Modelica.Math.Distributions.Interfaces.partialQuantile (Common interface of quantile functions (= inverse cumulative distribution functions)).

Inputs

Outputs