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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t0943601.png" />. The truncated distribution corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t0943602.png" /> is understood to be the distribution function
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$#C+1 = 24 : ~/encyclopedia/old_files/data/T094/T.0904360 Truncated distribution
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t0943603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
In the particular case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t0943604.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t0943605.png" />) the truncated distribution is said to be right truncated (left truncated).
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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
 
Together with (1) one considers truncated distribution functions of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t0943606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \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 .$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t0943607.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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In (1) the mass concentrated outside  $  [ a, b] $
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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 $,
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one usually takes for  $  c $
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the point  $  c = 0 $),
 +
and in (3) this mass is located at the extreme points  $  a $
 +
and  $  b $.
  
In (1) the mass concentrated outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t0943608.png" /> is distributed over the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t0943609.png" />, in (2) it is located at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436010.png" /> (in this case, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436011.png" />, one usually takes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436012.png" /> the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436013.png" />), and in (3) this mass is located at the extreme points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436015.png" />.
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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 $.
  
A truncated distribution of the form (1) may be interpreted as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436016.png" /> be a random variable with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436017.png" />. Then the truncated distribution coincides with the conditional distribution of the random variable under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436018.png" />.
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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
  
The concept of a truncated distribution is closely connected with the concept of a truncated random variable: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436019.png" /> 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}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436020.png" /></td> </tr></table>
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\right .$$
  
The distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436021.png" /> is a truncated distribution of type (3) (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436023.png" />) with respect to the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094360/t09436024.png" />.
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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.
 
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.

Latest revision as of 14:56, 7 June 2020


A 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

[1] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)
[2] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[3] W. Feller, "An introduction to probability theory and its applications", 1–2 , Wiley (1957–1971)
[4] M. Loève, "Probability theory" , Springer (1977)
How to Cite This Entry:
Truncated distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Truncated_distribution&oldid=49484
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article