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A [[Probability distribution|probability distribution]] obtained from a given distribution by transfer of probability mass outside a given interval to within this interval. Let a probability distribution on the line be given by a distribution function  $  F $. 
The truncated distribution corresponding to  $  F $
is understood to be the distribution function

$$ \tag{1 }
F _ {a,b} ( x)  = \ 
\left \{

\begin{array}{ll}
 0  &\textrm{ for }  x \leq  a,  \\

\frac{F ( x) - F ( a) }{F ( b) - F ( a) }
  &\textrm{ for }  a < x \leq  b,  \\
 1  &\textrm{ for }  x > b, a < b.  \\
\end{array}

 \right .$$

In the particular case  $  a = - \infty $(
$  b = \infty $) 
the truncated distribution is said to be right truncated (left truncated).

Together with (1) one considers truncated distribution functions of the form

$$ \tag{2 }
F _ {a,b} ( x)  = \ 
\left \{

\begin{array}{ll}
 0  &\textrm{ for }  x \leq  a,  \\
F ( x) - F ( a)  &\textrm{ for }  a < x < c,  \\
F ( x) + 1 - F ( b)  &\textrm{ for }  c \leq  x < b,  \\
 1  &\textrm{ for }  x \geq  b,  \\
\end{array}

 \right .$$

$$ \tag{3 }
F _ {a,b} ( x)  =  \left \{ 
\begin{array}{ll}
 0  &\textrm{ for }  x < a,  \\
F ( x)  &\textrm{ for }  a \leq  x < b,  \\
 1  &\textrm{ for }  x \geq  b.  \\
\end{array}

 \right .$$

In (1) the mass concentrated outside  $  [ a, b] $
is distributed over the whole of  $  [ a, b] $, 
in (2) it is located at the point  $  c \in ( a, b] $(
in this case, when  $  a < 0 < b $, 
one usually takes for  $  c $
the point  $  c = 0 $), 
and in (3) this mass is located at the extreme points  $  a $
and  $  b $.

A truncated distribution of the form (1) may be interpreted as follows. Let  $  X $
be a random variable with distribution function  $  F $. 
Then the truncated distribution coincides with the conditional distribution of the random variable under the condition  $  a < X \leq  b $.

The concept of a truncated distribution is closely connected with the concept of a truncated random variable: If  $  X $
is a random variable, then by a truncated random variable one understands the variable

$$ 
X  ^ {c}  =  \left \{ 
\begin{array}{lll}
 X  &\textrm{ if }  &| X | \leq  c,  \\
 0  &\textrm{ if }  &| X | > c.  \\
\end{array}

 \right .$$

The distribution of  $  X  ^ {c} $
is a truncated distribution of type (3) (with  $  a=- c $, 
$  b= c $) 
with respect to the distribution of  $  X $.

The truncation operation — passing to the truncated distribution or truncated random variable — is a very widespread technical device. It makes it possible, by a minor change in the initial distribution, to obtain an analytic property — existence of all moments.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,   Yu.A. Rozanov,   "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1–2''' , Wiley  (1957–1971)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  M. Loève,   "Probability theory" , Springer  (1977)</TD></TR></table>