Difference between revisions of "Truncated distribution"
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| − | In the particular case | + | 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 | ||
| + | |||
| + | $$ \tag{1 } | ||
| + | F _ {a,b} ( x) = \ | ||
| + | \left \{ | ||
| + | |||
| + | In the particular case $ a = - \infty $( | ||
| + | $ b = \infty $) | ||
| + | the truncated distribution is said to be right truncated (left truncated). | ||
Together with (1) one considers truncated distribution functions of the form | Together with (1) one considers truncated distribution functions of the form | ||
| − | + | $$ \tag{2 } | |
| + | F _ {a,b} ( x) = \ | ||
| + | \left \{ | ||
| − | + | $$ \tag{3 } | |
| + | F _ {a,b} ( x) = \left \{ | ||
| − | In (1) the mass concentrated outside | + | In (1) the mass concentrated outside $ [ a, b] $ |
| + | is distributed over the whole of $ [ a, b] $, | ||
| + | in (2) it is located at the point $ c \in ( a, b] $( | ||
| + | in this case, when $ a < 0 < b $, | ||
| + | one usually takes for $ c $ | ||
| + | the point $ c = 0 $), | ||
| + | and in (3) this mass is located at the extreme points $ a $ | ||
| + | and $ b $. | ||
| − | A truncated distribution of the form (1) may be interpreted as follows. Let | + | A truncated distribution of the form (1) may be interpreted as follows. Let $ X $ |
| + | be a random variable with distribution function $ F $. | ||
| + | Then the truncated distribution coincides with the conditional distribution of the random variable under the condition $ a < X \leq b $. | ||
| − | The concept of a truncated distribution is closely connected with the concept of a truncated random variable: If | + | The concept of a truncated distribution is closely connected with the concept of a truncated random variable: If $ X $ |
| + | is a random variable, then by a truncated random variable one understands the variable | ||
| − | + | $$ | |
| + | X ^ {c} = \left \{ | ||
| − | The distribution of | + | 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. | ||
Revision as of 08:26, 6 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
$$ \tag{1 } F _ {a,b} ( x) = \ \left \{ In the particular case $ a = - \infty $( $ b = \infty $) the truncated distribution is said to be right truncated (left truncated). Together with (1) one considers truncated distribution functions of the form $$ \tag{2 } F _ {a,b} ( x) = \ \left \{
$$ \tag{3 } F _ {a,b} ( x) = \left \{ In (1) the mass concentrated outside $ [ a, b] $ is distributed over the whole of $ [ a, b] $, in (2) it is located at the point $ c \in ( a, b] $( in this case, when $ a < 0 < b $, one usually takes for $ c $ the point $ c = 0 $), and in (3) this mass is located at the extreme points $ a $ and $ b $. A truncated distribution of the form (1) may be interpreted as follows. Let $ X $ be a random variable with distribution function $ F $. Then the truncated distribution coincides with the conditional distribution of the random variable under the condition $ a < X \leq b $. The concept of a truncated distribution is closely connected with the concept of a truncated random variable: If $ X $ is a random variable, then by a truncated random variable one understands the variable $$ X ^ {c} = \left \{
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=49043
Truncated distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Truncated_distribution&oldid=49043
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article