# HistogramDistribution

HistogramDistribution[{x1,x2,}]

represents the probability distribution corresponding to a histogram of the data values xi.

HistogramDistribution[{{x1,y1,},{x2,y2,},}]

represents a multivariate histogram distribution based on data values {xi,yi,}.

HistogramDistribution[,bspec]

represents a histogram distribution with bins specified by bspec.

# Details  • HistogramDistribution returns a DataDistribution object that can be used like any other probability distribution.
• The probability density function for HistogramDistribution for a value is given by where is the number of data points in bin , is the width of bin , are bin delimiters, and is the total number of data points.
• The width of each bin is computed according to the values xi, the width according to the yi, etc.
• The following bin specifications bspec can be given:
•  n use n bins {w} use bins of width w {min,max,w} use bins of width w from min to max {{b1,b2,…}} use bins [b1,b2),[b2,b3),… Automatic determine bin widths automatically "name" use a named binning method fw apply fw to get an explicit bin specification {b1,b2,…} {xspec,yspec,…} give different x, y, etc. specifications
• Possible named binning methods include:
•  "FreedmanDiaconis" twice the interquartile range divided by the cube root of sample size "Knuth" balance likelihood and prior probability of a piecewise uniform model "Scott" asymptotically minimize the mean square error "Sturges" compute the number of bins based on the length of data "Wand" one-level recursive approximate Wand binning
• The probability density for value in a histogram distribution is a piecewise constant function.
• HistogramDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Examples

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## Basic Examples(2)

Create a histogram distribution of univariate data:

Use the resulting distribution to perform analysis, including visualizing distribution functions:

Compute moments and quantiles:

Create a histogram distribution of bivariate data:

Visualize the PDF and CDF:

Compute covariance and general moments:

## Scope(29)

### Basic Uses(5)

Create a distribution from a histogram of some data:

Compute probabilities from the distribution:

Create histogram distributions from quantity data:

Find select descriptive statistics:

Decrease the number of bins to decrease local sensitivity:

Increase the bin width to decrease local sensitivity:

Create distributions from histograms in higher dimensions:

Plot the univariate marginal PDFs:

Plot the bivariate marginal PDFs:

### Distribution Properties(10)

Estimate distribution functions:

Compute moments of the distribution:

Special moments:

General moments:

Quantile function:

Special quantile values:

Generate random numbers:

Compare with HistogramDistribution:

Compute probabilities and expectations:

Generating functions:

Estimate distribution functions for bivariate data:

Compute moments of a bivariate distribution:

Special moments:

General moments:

Generate random numbers:

Show the point distribution:

Having fewer bins yields a coarser approximation to the underlying distribution:

Generating functions:

### Binning(14)

Automatically compute the number of bins:

More data yields smaller bins:

Explicitly specify the number of bins to use:

Specify 5 and 50 bins, respectively:

Explicitly specify bin width:

Use bin widths 1. and 0.1:

Specify bin range and bin width:

Use bin widths of 1.5 and .15 respectively over fixed interval:

Provide explicit bin delimiters:

Use different automatic binning methods:

Delimit bins on integer boundaries using a binning function:

Automatically compute the number of bins for bivariate data:

More data yields smaller bins:

Explicitly specify the number of bins to use:

Explicitly specify bin width:

Specify bin range and bin width:

Explicitly give bin delimiters:

Use different automatic binning methods:

Use different bin specifications in each dimension:

Specify 3 bins in the row dimension and bin width 0.5 in the column dimension:

## Applications(6)

Compare an estimated density to a theoretical model:

Distribution of lengths of human chromosomes:

Compute the probability that the sequence length is greater than 15:

Compare the distributions of word length for some of the parts of speech:

The expected number of characters for a randomly chosen English noun:

Estimate the distribution of day-to-day point changes in the S&P 500 index:

Compute the probability of a 1% point change or more on a given day:

Determine the number of bins to use for bimodal data by Knuth's Bayesian method:

The optimal number of bins maximizes the log of the posterior density:

Density estimates using Knuth's method, Scott's rule, and the FreedmanDiaconis rule:

Knuth's method outperforms the other two in terms of LogLikelihood:

Construct a continuous version of the empirical cumulative distribution function:

Cumulative distribution function for HistogramDistribution is piecewise linear:

Compute Cramervon Mises distance between the two distributions:

## Properties & Relations(10)

The PDF of HistogramDistribution is equivalent to a probability density Histogram:

The resulting density estimate integrates to unity:

The precision of the output matches the precision of the data:

The PDF is piecewise constant:

The CDF and SurvivalFunction are piecewise linear:

The HazardFunction is linear fractional:

HistogramDistribution is a MixtureDistribution of uniform distributions:

HistogramDistribution is a consistent estimator of the underlying distribution:

HistogramDistribution works with the values only when the input is a TimeSeries:

Compare to the histogram distribution of the values:

HistogramDistribution works with all the values together when the input is a TemporalData:

Compare to the histogram distribution calculated with the values from all the paths:

## Possible Issues(1)

It is possible to drop data from the estimation by specifying a binning range:

Specifying a width alone uses all the data:

## Neat Examples(1)

Random pop art with HistogramDistribution: