In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a
normally distributed random variable by bounding the random variable from either below or above (or both). The
truncated normal distribution has wide applications in statistics and
econometrics
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. ...
.
Definitions
Suppose
has a normal distribution with mean
and variance
and lies within the interval
. Then
conditional on
has a truncated normal distribution.
Its
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
,
, for
, is given by
:
and by
otherwise.
Here,
:
is the probability density function of the
standard normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
and
is its
cumulative distribution function
:
By definition, if
, then
, and similarly, if
, then
.
The above formulae show that when