Difference between revisions of "Truncated distribution"
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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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− | + | <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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− | + | 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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− | In the particular case | ||
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− | 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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− | + | <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 | + | 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" />. |
− | is distributed over the whole of | ||
− | in (2) it is located at the point | ||
− | in this case, when | ||
− | one usually takes for | ||
− | the point | ||
− | and in (3) this mass is located at the extreme points | ||
− | and | ||
− | 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 <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" />. |
− | be a random variable with distribution function | ||
− | Then the truncated distribution coincides with the conditional distribution of the random variable under the condition | ||
− | 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 <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 |
− | is a random variable, then by a truncated random variable one understands the variable | ||
− | + | <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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− | The distribution of | + | 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" />. |
− | is a truncated distribution of type (3) (with | ||
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− | with respect to the distribution of | ||
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 14:53, 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 . The truncated distribution corresponding to
is understood to be the distribution function
![]() | (1) |
In the particular case (
) the truncated distribution is said to be right truncated (left truncated).
Together with (1) one considers truncated distribution functions of the form
![]() | (2) |
![]() | (3) |
In (1) the mass concentrated outside is distributed over the whole of
, in (2) it is located at the point
(in this case, when
, one usually takes for
the point
), and in (3) this mass is located at the extreme points
and
.
A truncated distribution of the form (1) may be interpreted as follows. Let be a random variable with distribution function
. Then the truncated distribution coincides with the conditional distribution of the random variable under the condition
.
The concept of a truncated distribution is closely connected with the concept of a truncated random variable: If is a random variable, then by a truncated random variable one understands the variable
![]() |
The distribution of is a truncated distribution of type (3) (with
,
) with respect to the distribution of
.
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) |
Truncated distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Truncated_distribution&oldid=49043