In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a truncated distribution is a
conditional distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the ...
that results from restricting the domain of some other
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
. 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.
Definition
The following discussion is in terms of a random variable having a
continuous distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
although the same ideas apply to
discrete distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s. 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
both of which have infinite
Support (mathematics), support. 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,
. That is to say, suppose we wish to know how
is distributed given
.
:
where
for all