Truncated Distribution¶

A probability law conditioned on a positive-probability retained region and renormalized over that region.

Version: v1 · 2026-10-03 · History
Domain-specific #: 13677
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Probability Theory, Statistics → Mathematics
Aliases: Conditional Truncated Distribution

Core Idea¶

A truncated distribution is a new probability law made from an existing one by keeping only values in a specified region and then rescaling their probabilities to total one. Suppose \(X\) follows a parent distribution and \(A\) is the retained event. The construction requires \(P(X\in A)>0\). For any event \(B\), the new law is \(P_T(B)=P(X\in B\cap A)/P(X\in A)\). This is more than crossing out out-of-range values: without the division, the remaining probabilities generally total less than one.[^{ref-b3adff7b19b6][}ref-b88337095092]

For example, keep only normally distributed values between two limits: the normal density inside the interval is divided by the normal probability of being in that interval. The same operation works for a Poisson count when zero is excluded: positive counts are divided by the chance of a nonzero count. The parent law and exact retained event matter; if a discrete distribution has probability at a boundary, whether the boundary is included changes the result.[^{ref-b3adff7b19b6][}ref-eecd377f5dc3]

Scope of Application¶

This is a probability-and-statistics construct, useful when a model deliberately limits possible draws or when a dataset contains only values that pass an inclusion rule. It is not confined to normal curves: continuous and discrete parent laws can both be truncated if the retained event has positive probability. SciPy documents a continuous truncation operation applicable beyond normals, and the actuar package documents the positive-only Poisson case.[^{ref-b88337095092][}ref-eecd377f5dc3]

An empirical selection rule must actually match the proposed conditional law. Censoring is different: a unit is still represented, but its exact value is incompletely known or limit-coded. Also, the mathematical definition does not require the analyst to be ignorant of how many units were excluded; that count may or may not be available from the study design.[^ref-b3adff7b19b6]

Clarity¶

The label prompts three concrete questions: What was the original distribution? Which outcomes remain? What was their total probability before truncation? These distinguish a genuine conditional law from a plot cropped to show a convenient range or a naturally bounded law with no specified wider parent.

It also separates the included group's properties from the whole population's properties. Stata's labor-hours example shows that a positive-hours sample can be the appropriate target for a question about workers, while its mean does not automatically answer a question about all women in the source population.[^ref-b3adff7b19b6]

Manages Complexity¶

Many excluded values can be summarized by a single probability: the chance that an original draw falls in the retained region. Dividing surviving probabilities by this number yields a complete normalized law. For a normal interval, that number is a difference of cumulative-normal probabilities; for a positive Poisson count it is \(1-e^{-\lambda}\). The shorter expression preserves the essential effect of selection without separately listing each excluded value.[^{ref-b3adff7b19b6][}ref-eecd377f5dc3]

The compression has a limit. The conditional law describes relative probabilities among included values; it does not, by itself, establish the excluded population size or prove that the assumed parent family was correct.

Abstract Reasoning¶

To test a proposed case, identify a parent probability law and a retained event with positive mass. Check that every excluded value has zero probability under the new law and that all surviving event probabilities are multiplied by the same reciprocal retained mass. If only support was shortened, with no rescaling, the result is not yet a probability distribution.

For data analysis, ask what was observed and what target is wanted. Positive-only counts may call for a zero-truncated count likelihood; boundary-coded values may instead call for a censoring model. The distinction follows from whether units outside the range are absent or still recorded in incomplete form, not from whether a graph has a visible cutoff.[^{ref-b3adff7b19b6][}ref-eecd377f5dc3]

Knowledge Transfer¶

The same conditional-law operation transfers literally across distribution families. A normal law restricted to an interval and a Poisson law restricted to positive integers use the identical retained-event/normalization structure, although their densities or masses differ. Beyond probability theory, cutting off part of an object instantiates the broader live prime Truncation, but a shortened string or cropped image does not generally renormalize probability.

[^ref-b3adff7b19b6]: StataCorp, Truncated regression [R] truncreg, official manual, pp. 3–6, conditional-normal equation, example and truncation/censoring distinction. [^ref-b88337095092]: SciPy developers, scipy.stats.truncate, v1.15.2 API definition and examples. [^ref-eecd377f5dc3]: Vincent Goulet and actuar package contributors, The Zero-Truncated Poisson Distribution, actuar reference manual, Details and Examples.

Relationships to Other Abstractions¶

Current abstraction Truncated Distribution Domain-specific

Parents (1) — more general patterns this builds on

Truncated Distribution is a kind of Probability Distribution Domain-specific

A truncated distribution is a probability law conditioned on a positive-probability retained event.

Neighborhood in Abstraction Space¶

Truncated Distribution sits in a sparse region of the domain-specific corpus (86^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Foundations of Probability & Inference (29 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08