# Truncated Distributions

The package provides the truncated function which creates the most appropriate distribution to represent a truncated version of a given distribution.

A truncated distribution can be constructed using the following signature:

Distributions.truncatedFunction
truncated(d0::UnivariateDistribution; [lower::Real], [upper::Real])
truncated(d0::UnivariateDistribution, lower::Real, upper::Real)

A truncated distribution d of a distribution d0 to the interval $[l, u]=$[lower, upper] has the probability density (mass) function:

$$$f(x; d_0, l, u) = \frac{f_{d_0}(x)}{P_{Z \sim d_0}(l \le Z \le u)}, \quad x \in [l, u],$$$

where $f_{d_0}(x)$ is the probability density (mass) function of $d_0$.

The function throws an error if $l > u$.

truncated(d0; lower=l)           # d0 left-truncated to the interval [l, Inf)
truncated(d0; upper=u)           # d0 right-truncated to the interval (-Inf, u]
truncated(d0; lower=l, upper=u)  # d0 truncated to the interval [l, u]
truncated(d0, l, u)              # d0 truncated to the interval [l, u]

The function falls back to constructing a Truncated wrapper.

Implementation

To implement a specialized truncated form for distributions of type D, one or more of the following methods should be implemented:

• truncated(d0::D, l::T, u::T) where {T <: Real}: interval-truncated
• truncated(d0::D, ::Nothing, u::Real): right-truncated
• truncated(d0::D, l::Real, u::Nothing): left-truncated
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In the general case, this will create a Truncated{typeof(d)} structure, defined as follows:

Distributions.TruncatedType
Truncated

Generic wrapper for a truncated distribution.

The truncated normal distribution is a particularly important one in the family of truncated distributions. Unlike the general case, truncated normal distributions support mean, mode, modes, var, std, and entropy.

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Many functions, including those for the evaluation of pdf and sampling, are defined for all truncated univariate distributions:

Functions to compute statistics, such as mean, mode, var, std, and entropy, are not available for generic truncated distributions. Generally, there are no easy ways to compute such quantities due to the complications incurred by truncation. However, these methods are supported for truncated normal distributions Truncated{<:Normal} which can be constructed with truncated(::Normal, ...).