This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 BUILT-IN MATHEMATICA SYMBOL

# BenfordDistribution

 BenfordDistribution[b] represents a Benford distribution with base parameter b.
• The probability for integer value in a Benford distribution is proportional to for , and is otherwise.
Probability density function:
Cumulative distribution function:
Mean:
Variance:
Median:
Probability density function:
 Out[1]=
 Out[2]=

Cumulative distribution function:
 Out[1]=
 Out[2]=

Mean:
 Out[1]=
 Out[2]=

Variance:
 Out[1]=
 Out[2]=

Median:
 Out[1]=
 Scope   (6)
Generate a set of pseudorandom numbers that are Benford distributed:
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 is defined for :
Kurtosis is defined for :
Hazard function:
Quantile function:
 Applications   (4)
CDF of BenfordDistribution is an example of a right continuous function:
Benford's distribution approximates distributions of values spanning multiple orders of magnitude. Consider a sample from a heavy-tailed distribution:
Find the order of magnitude between minimum and maximum:
Extract first digits:
Compare the histogram with the PDF of the corresponding BenfordDistribution:
Now consider a sample from a light-tailed distribution:
Find the order of magnitude between minimum and maximum:
Compare the histogram with the PDF of the corresponding BenfordDistribution:
Check whether the population of the largest cities in the United States follows Benford distribution:
The population of the 100 largest cities does not follow Benford distribution very well:
Consider physical constants:
Find the first digits, not taking units into account:
The first digits are not uniformly distributed; more likely their distribution follows Benford's law:
New in 8