# Truncated distribution

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) |

**How to Cite This Entry:**

Truncated distribution.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Truncated_distribution&oldid=25538