# TruncatedDistribution

TruncatedDistribution[{xmin,xmax},dist]

represents the distribution obtained by truncating the values of dist to lie between xmin and xmax.

TruncatedDistribution[{{xmin,xmax},{ymin,ymax},},dist]

represents the distribution obtained by truncating the values of the multivariate distribution dist to lie between xmin and xmax, ymin and ymax, etc.

# Details

• The probability density for TruncatedDistribution[{xmin,xmax},dist] is given by for , where is the PDF and is the CDF of dist, and is zero otherwise.
• Common cases for {xmin,xmax} include:
•  {-∞,xmax} truncated from above {xmin,∞} truncated from below {xmin,xmax} doubly truncated {-∞,∞},None no truncation
• TruncatedDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• TruncatedDistribution[{xmin,xmax},dist] represents a statistical distribution modeling data that is a constant multiple of the univariate distribution dist for all in the interval and that is constantly 0 for . The terms non-truncated, truncated from below, truncated from above, and doubly truncated are used to describe univariate truncations for which {xmin,xmax} has the form {-,}, {xmin,}, {-,xmax}, and {xmin,xmax}, respectively, while univariate dist may be either continuous (e.g. NormalDistribution, GammaDistribution, or BetaDistribution) or discrete (e.g. PoissonDistribution, BinomialDistribution, or BernoulliDistribution) and may be defined in terms of transformations, censoring, or truncations (by way of TransformedDistribution, CensoredDistribution, and TruncatedDistribution, respectively) of known distributions.
• The multivariate TruncatedDistribution[{{,}, ,{,}},dist] is defined analogously and thus represents the distribution of vectors taken from the multivariate distribution dist and whose component is truncated to lie in the interval . As in the univariate case, multivariate dist may again be either continuous (e.g. MultinormalDistribution) or discrete (e.g. MultivariateHypergeometricDistribution), and may also be defined as a copula or product (using CopulaDistribution and ProductDistribution, respectively) of known distributions.
• Truncated distributions arise when datasets contain values that lie outside an interval considered "acceptable". For example, if the first people to file income taxes were to be audited from a pool of taxpayers, then any statistical analysis performed on the audited returns would come in the form of truncated data, provided that no information was stored regarding the taxpayers who avoided audits. Such data is common in fields such as survival analysis, finance, actuarial science, and economics, and a variety of specialized statistical tools (e.g. truncated regression) exists to analyze such datasets.
• By definition, CensoredDistribution[{xmin,xmax},dist] is equivalent to TransformedDistribution[g,xdist], where g is given by Piecewise[{{0,x<=xmin},{h,xmin<x<xmax},{0,x>=xmax}}] for , where (respectively, ) is the PDF (respectively, the CDF) of dist. TruncatedDistribution is often confused with CensoredDistribution, though the two are fundamentally different in the sense that truncation distributes the probability over the truncation interval, while censoring puts the full probability at the end of the censoring interval.

# Examples

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## Basic Examples(3)

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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