In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution.
Truncated distributions arise in practical statistics in cases where the ability to record, or even to know about, occurrences is limited to values which lie above or below a given threshold or within a specified range.
For example, if the dates of birth of children in a school are examined, these would typically be subject to truncation relative to those of all children in the area given that the school accepts only children in a given age range on a specific date.
There would be no information about how many children in the locality had dates of birth before or after the school's cutoff dates if only a direct approach to the school were used to obtain information.
Where sampling is such as to retain knowledge of items that fall outside the required range, without recording the actual values, this is known as censoring, as opposed to the truncation here.
[1] The following discussion is in terms of a random variable having a continuous distribution although the same ideas apply to discrete distributions.
Similarly, the discussion assumes that truncation is to a semi-open interval y ∈ (a,b] but other possibilities can be handled straightforwardly.
Suppose we have a random variable,
that is distributed according to some probability density function,
, with cumulative distribution function
Suppose we wish to know the probability density of the random variable after restricting the support to be between two constants so that the support,
Note that the denominator in the truncated distribution is constant with respect to the
is a density: Truncated distributions need not have parts removed from the top and bottom.
is the cumulative distribution function.
is the cumulative distribution function.
Suppose we wish to find the expected value of a random variable distributed according to the density
The expectation of a truncated random variable is thus: where again
be the lower and upper limits respectively of support for the original density function
(which we assume is continuous), properties of
is some continuous function with a continuous derivative, include: Provided that the limits exist, that is:
The truncated normal distribution is an important example.
[2] The Tobit model employs truncated distributions.
Suppose we have the following set up: a truncation value,
, is selected at random from a density,
, is selected at random from the truncated distribution,
and wish to update our belief about the density of
, we set a lower bound of
are the unconditional density and unconditional cumulative distribution function, respectively.
By Bayes' rule, which expands to Suppose we know that t is uniformly distributed from [0,T] and x|t is distributed uniformly on [0,t].
Let g(t) and f(x|t) be the densities that describe t and x respectively.