# Truncated Distributions¶

The package provides a type, named Truncated, to represented truncated distributions, which is defined as below:

immutable Truncated{D<:UnivariateDistribution,S<:ValueSupport} <: Distribution{Univariate,S}
untruncated::D      # the original distribution (untruncated)
lower::Float64      # lower bound
upper::Float64      # upper bound
lcdf::Float64       # cdf of lower bound
ucdf::Float64       # cdf of upper bound

tp::Float64         # the probability of the truncated part, i.e. ucdf - lcdf
logtp::Float64      # log(tp), i.e. log(ucdf - lcdf)
end


A truncated distribution can be constructed using the constructor Truncated as follows:

Truncated(d, l, u):

Construct a truncated distribution.

Parameters: d – The original distribution. l – The lower bound of the truncation, which can be a finite value or -Inf. u – The upper bound of the truncation, which can be a finite value of Inf.

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

• maximum
• minimum
• insupport
• pdf
• logpdf
• cdf
• logcdf
• ccdf
• logccdf
• quantile
• cquantile
• invlogcdf
• invlogccdf
• rand
• rand!
• median

However, 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.

## Truncated Normal Distribution¶

The truncated normal distribution is a particularly important one in the family of truncated distributions. We provide additional support for this type.

One can construct a truncated normal distribution using the common constructor Truncated, as

Truncated(Normal(mu, sigma), l, u)


or using a dedicated constructor TruncatedNormal as

TruncatedNormal(mu, sigma, l, u)


Also, we provide additional methods to compute various statistics for truncated normal:

• mean
• mode
• modes
• var
• std
• entropy