This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)

BenktanderGibratDistribution

BenktanderGibratDistribution
represents a Benktander distribution of type I with parameters a and b.
  • The probability density for value in a Benktander-Gibrat distribution is proportional to for .
Probability density function:
Cumulative distribution function:
Mean and variance:
The median can be found numerically:
Probability density function:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
In[3]:=
Click for copyable input
Out[3]=
 
Cumulative distribution function:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
In[3]:=
Click for copyable input
Out[3]=
 
Mean and variance:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
 
The median can be found numerically:
In[1]:=
Click for copyable input
Out[1]=
Generate a set of pseudorandom numbers that have Benktander-Gibrat distribution:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare the density histogram of the sample with the PDF of the estimated distribution:
Skewness:
Kurtosis:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Hazard function:
Quantile function:
An insurance company finds its claim sizes in units of deductibles follow a Benktander type I distribution with parameters and . Find the probability that claim sizes will exceed 2:
Compute the mean excess function for a Benktander type I distribution:
For large it approaches that of log-normal, also known as Gibrat distribution:
Now replace the complimentary error function with its asymptotics at large arguments:
Find stationary renewal distribution associated with a Benktander type I distribution:
Compare it with BeniniDistribution:
Parameter influence on the CDF for each :
Benktander type I distribution is subexponential:
Relationships to other distributions:
New in 8