Contents - Index

Pearson Family of Distributions

Shape:

The Pearson family of distributions is made up of seven distributions: Type I-VII.  It covers any specified average, standard deviation, skewness and kurtosis.  Together they form a 4-parameter family of distributions that covers the entire skewness-kurtosis region other than the impossible region.

       

The seven types are described below.
  • Type I: Beta Distribution
  • Type II: Special case of beta distribution that is symmetrical
  • Type III: Gamma Distribution
  • Type IV: Region above Type V
  • Type V: 3 parameter distribution represented by curve
  • Type VI: Region between Gamma and Type V
  • Type VII: Special case of Type IV that is symmetrical

      The special cases can be ignored (II and VII).  Type I and III and are alias for distribution already covered.  That leaves Type IV, V and VI as new distributions.  

    Density Function - Type IV Distribution:

    The density function of the Type IV distribution is shown below:

           

    The equation, parameters and bounds of the density function are:

            

    Moments - Type IV Distribution:

    The moments of the Type IV distribution can be calculated from the parameters as shown below:

           

    Density Function - Type V Distribution (Positive Skewness):

    The density function of the Type V distribution is shown below:

           

    The equation, parameters and bounds of the density function are:

            

    Moments - Type V Distribution (Positive Skewness):

    The moments of the Type V distribution can be calculated from the parameters as shown below:

           

    Density Function - Type VI Distribution:

    The density function of the Type VI distribution is shown below:

           

    The equation, parameters and bounds of the density function are:

            

    Moments - Type VI Distribution:

    The moments of the Type VI distribution can be calculated from the parameters as shown below: