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 $ 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)