Distributions

Standard distributions

Most of the following distributions are listed in the first GUM supplement. The algorithms how to generate random values of these distributions using numbers from a random number generator for rectangular distributed values are also listed there.

Please note that most parameters in the following examples of the influences are parametrised.

Normal/Gauss distribution

Example:

<distribution>
  <gauss>
    <mu>0</mu>
    <sigma>1</sigma>
  </gauss>
</distribution>

Result:

Rectangular distribution

Example:

<distribution>
  <rectangle>
    <mean>0</mean>
    <width>0.5</width>
  </rectangle>
</distribution>

Result:

Triangular distribution

Example:

<distribution>
  <triangle>
    <mean>0</mean>
    <width>0.5</width>
  </triangle>
</distribution>

Result:

Trapezoidal distribution

Example:

<distribution>
  <trapez>
    <lower>4</lower>
    <upper>8</upper>
    <beta>2</beta>
  </trapez>
</distribution>

Result:

Warning:

Earlier versions of MUSE up to version 0.6.2 used an inverse beta for definition in tag__ <beta>__. To be consistent to GS1 this changes with version 0.6.3 to the definition given there.</beta>

Curvelinear trapezoidal distribution

Example:

<distribution>
  <cltrapez>
    <lower>4</lower>
    <upper>8</upper>
    <inexactness>1</inexactness>
  </cltrapez>
</distribution>

Result:

Arcsine distribution

Example:

<distribution>
  <arcsine>
    <lower>4</lower>
    <upper>8</upper>
  </arcsine>
</distribution>

Result:

Exponential distribution

Example:

<distribution>
  <exponential>
    <lambda parameter="#ExponentialLambda"/>
  </exponential>
</distribution>

Gamma distribution

Example:

<distribution>
  <gamma>
    <alpha parameter="#GammaAlpha"/>
    <beta parameter="#GammaBeta"/>
  </gamma>
</distribution>

Student-t distribution

Example 1: Precalculated values

<distribution>
  <studentt>
    <xbar parameter="#StudentTxBar"/>
    <std parameter="#StudentTStd"/>
    <dgf parameter="#StudentTDf"/>
  </studentt>
</distribution>

Example 2: Providing data

<distribution>
  <studentt>
    <values>
      <value>2.0</value>
      <value>3.0</value>
      <value>4.0</value>
      <value>9.0</value>
    </values>
  </studentt>
</distribution>

Special distributions

Constant values

Example:

<distribution>
  <constant>
    <value parameter="#ValueConstant"/>
  </constant>
</distribution>

Density distribution: Presampled values

You might get to the problem, that you need a distribution that is not yet supported by MUSE. We have a very rich solution for that. MUSE is able to handle files with density definitions of distributions and interpret them as a given distribution. The file consist of a list of numbers in different formats and takes this numbers as random numbers. The definition of such a distribution looks like this:

<distribution>
   <density startvalue="1000" filetype="binary" dimension="2">
     <filename>path/density.bin</filename>
   </density>
</distribution>

The distribution will use the given file to get random numbers. It starts with the number on position of startvalue or else on a random position. Each time when the simulation system askes for a new random number the distribution returns the next value in the file. If the distribution reaches the value on the last position, it resets the pointer to the first position and continues from there on. By filetype the format of the file is specified. At the moment we support this formats:

• plain text: Output file of MUSE without additional parameters
• binary: Output file of MUSE with parameter -b
• linesep: Each value is separated by a line break
• openBugs: Output file of Bugs-software

The parameter dimension is only neccessary for \Muse files. It tells the programm which dimension or set of values to use, if there is more than one dimension stored in the data file.

Complex distribution

You can find a detailed description of this distribution Complex valued quantities.

Correlated normal distribution

Related Topics

• [Cut distributions]