and everywhere else. Notice that has the same support as .There is, unfortunately, an ambiguity about the term Truncated Distribution. When one refers to a truncated distribution they could be referring to where one has removed the parts from the distribution but not scaled up the distribution, or they could be referring to the . g(x) is not a generally a probability density function since it does not integrate to one, where as is a probability density function. In this article, a truncated distribution refers to
Notice that in fact is a distribution::.
Truncated distributions need not have parts removed from the top and bottom. A truncated distribution where the just bottom of the distribution has been removed is as follows:
:
where for all and everywhere else, and is the cumulative distribution function.
A truncated distribution where the top of the distribution has been removed is as follows:
:
where for all and everywhere else, and is the cumulative distribution function.
Expectation of truncated random variable
Suppose we wish to find the expected value of a random variable distributed according to the density and a cumulative distribution of given that the random variable, , is greater than some known value . The expectation of a truncated random variable is thus:
where again is again for all and everywhere else.
Letting and be the lower and upper limits respectively of support for (i.e. the original density) properties of where is some continuous function of with a continuous derivative and where is assumed continuous include:
(i)
(ii)
(iii)
(iv)
(v)
Provided that the limits exist, that is: , and where represents either or .
The Tobit model employs truncated distributions.
Random truncation
Suppose we have the following set up: a truncation value, , is selected at random from a density, , but this value is not observed. Then a value, , is selected at random from the truncated distribution, . Suppose we observe and wish to update our belief about the density of given the observation.
First, by definition:
, and
Notice that must be greater than , hence when we integrate over , we set a lower bound of . and are the unconditional density and unconditional cumulative distribution function, respectively.
By Bayes Rule:
which expands to:
Two uniform distributions (example)
Suppose we know that t is uniformly distributed from [0,T] and x|t is distributed uniformly from [0,t] . Let g(t) and f(x|t) be the densities that describe t and x respectively. Suppose we observe a value of x and wish to know the distribution of t given that value of x.
ee also
*Truncated normal distribution
*Truncation (statistics)
*Truncated mean