This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# TruncatedDistribution

 TruncatedDistribution represents the distribution obtained by truncating the values of dist to lie between and . TruncatedDistributionrepresents the distribution obtained by truncating the values of the multivariate distribution dist to lie between and , and , etc.
• The probability density for TruncatedDistribution is given by for , where is the PDF and is the CDF of dist, and is zero otherwise.
• Common cases for include:
 {-∞,xmax} truncated from above {xmin,∞} truncated from below {xmin,xmax} doubly truncated {-∞,∞},None no truncation
Simple truncated distributions:
Define a truncated univariate distribution:
Define a truncated multivariate distribution:
Simple truncated distributions:
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Define a truncated univariate distribution:
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Define a truncated multivariate distribution:
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 Scope   (34)
Define various truncations for a univariate continuous distribution:
The resulting PDF is 0 outside the truncation region:
Define various truncations for a univariate discrete distribution:
The left endpoint is not included, but the right endpoint is included:
Define a right-truncated distribution:
Compare PDFs:
Define a left-truncated distribution:
Compare probability density functions:
Compare means:
Compare standard deviations:
Define a doubly truncated distribution:
Compare means:
Compare variances:
Define a truncated multivariate continuous distribution:
Compute the expectation of an expression for this distribution:
Find a few moments:
Define a truncated multivariate discrete distribution:
Compare probabilities for a point outside the truncation region:
Generate a random sample:
Compare the means:
Define a truncated multivariate discrete distribution:
Compare skewness:
Compare kurtosis:
Estimate a truncation interval using EstimatedDistribution:
Fit a truncated normal:
Define a left-truncated continuous distribution:
Compare the probability density functions:
Cumulative distribution function of the truncated exponential distribution:
Statistical measures:
Compare to the original distribution:
Define a right-truncated discrete distribution:
Compare the PDFs:
The truncated distribution is the same as the following:
Define a truncation of a UniformDistribution:
Probability density function:
Compare to the uniform distribution defined on the truncation interval:
Define a truncation of a DiscreteUniformDistribution:
Probability density function:
Compare to the uniform distribution defined on the truncation interval:
Truncation does not include the left endpoint, hence the resulting discrete distribution:
Define a truncated binormal distribution:
Compare the PDFs for the binormal distribution and the truncated version:
Probability density function of the truncated binormal:
Characteristic function:
Define a discrete multivariate truncated distribution:
Perform statistical operations on this distribution:
Generate a set of pseudorandom numbers from a truncated distribution:
Compare the histogram to the PDF:
Compare probability density functions:
Define a truncated EmpiricalDistribution:
Compare cumulative distribution functions:
Define a truncated HistogramDistribution:
Probability density function:
Define a truncated TruncatedDistribution:
Find a probability density function:
Identify as a truncated distribution:
Define a truncated MixtureDistribution:
Compare probability density functions:
Define a truncated OrderDistribution:
Find the probability that the maximum of a Poisson sample is greater than 6, assuming it is greater than 5:
Find the probability that the maximum is greater than 6 without assuming it is greater than 5:
Define a truncated TransformedDistribution:
Compare with the transformation of the truncated normal:
Define a truncated ParameterMixtureDistribution:
Compare PDFs:
Find the probabilities of most-likely values for both distributions:
Define a truncated ProductDistribution:
Compare the probability density functions:
Compare with the PDF of the product of the truncated distributions:
Define a truncated MarginalDistribution:
Compare the probability density functions:
Define a truncated CensoredDistribution:
Compare the probability density functions:
GumbelDistribution truncated to a positive axis follows a GompertzMakehamDistribution:
NormalDistribution truncated to a positive axis follows a HalfNormalDistribution:
ParetoDistribution is closed under truncation:
UniformDistribution is closed under truncation:
DiscreteUniformDistribution is closed under truncation:
ZipfDistribution is closed under truncation:
 Applications   (5)
A grocery store orders pounds of produce for price per pound to be sold during the day. It sells the produce with margin per pound. The amount of produce sold in a day follows some distribution . The unsold produce is discarded at the end of the day. Compute so that it maximizes the daily profit:
Assuming 30% margin, and using LogNormalDistribution for distribution of demand:
The diameter of an American cranberry follows a normal distribution with mean 16 mm and standard deviation 1.6 mm. A fruit must be at least 15 mm across to be sold as whole; otherwise it is used in the production of cranberry sauce. Find the size distribution of the fruits being sold as whole:
Compare probability density functions:
Find the average diameter of sold fruits:
The probability that a sold fruit is at least 18 mm in diameter:
Truncated distribution can be used to control display of long tail distributions. Consider a sample:
Fit a Pareto distribution to the data:
Compare the histogram of the sample with the PDF of the estimated distribution:
Due to the long tail, the histogram range has to be adjusted and the distribution truncated:
Consider the width of certain species of crab:
Fit a DagumDistribution to the data:
Compare the histogram to the PDF of the estimated distribution:
Usually the dimensions of the caught crab species fall in a certain range:
Fit a truncated Dagum distribution to the data:
Compare log-likelihood values to see that the fit with a truncated distribution is better:
A company manufactures nails with length normally distributed and a mean of 0.5 inches. Given that the length of 50% of the produced nails differs less than 0.05 inches from the mean, find the standard deviation:
You then need to solve:
Plot to find the approximate value:
The standard deviation:
Truncating a distribution is equivalent to conditioning on an interval:
The PDF of a truncated distribution has nonzero values only inside the truncation interval:
Compare the density functions:
Construct the PDF of a truncated distribution by using properties of the underlying distribution:
Compare censoring with truncating for a discrete distribution:
While truncating, the weight from outside is evenly distributed over the truncation interval:
While censoring, the weight from outside is placed at the ends of the censoring interval:
Compare censoring and truncating of a continuous distribution:
While truncating, the probability is distributed over the truncation interval:
While censoring, the probability is put at the end of the censoring interval:
Double truncation of a bivariate distribution:
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