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)