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 the probability law obtained by restricting a specified parent law to a retained set of values and renormalizing the probability within that set. If a random variable \(X\) has parent law \(P_X\), and \(A\) is the retained event with \(P_X(A)>0\), then the truncated law assigns any event \(B\) probability \(P_T(B)=P_X(B\cap A)/P_X(A)\). Outcomes outside \(A\) receive zero under this new law; probabilities inside it are rescaled to total one. This is a construction on a whole law, not merely the act of deleting rows from a table. Stata's conditional truncated-normal density and SciPy's general continuous truncation operation both instantiate the construction.^[1]^[2]

For a continuous density \(f\) retained on \(a<X<b\), the density inside that interval is \(f(x)/[F(b)-F(a)]\) and zero outside, provided the denominator is positive. For a discrete law, the analogous point masses are divided by the probability of the retained set. Endpoint convention matters when the parent law has atoms: \(F(b)-F(a)\) is the mass of \(a<X\le b\), whereas retaining \(a\le X\le b\) can require adding the mass at \(a\). The general set formula avoids hiding this distinction.^[1]^[3]

Truncation is frequently motivated by data selection: only observations in a particular range enter a dataset. But the conditional law is mathematically meaningful without an empirical exclusion process—for example, when a simulator deliberately samples from a constrained range. Censoring is different: units remain represented while their exact values are only partly known or recorded at a limit. Stata explicitly contrasts the renormalized truncated law with its censored model, which places probability from censored regions at boundary reports.^[1]

Structural Signature¶

Sig role-phrases: parent probability law → declared retained event → positive retained mass → conditional reallocation → target-sensitive interpretation.

Parent probability law. The original distribution determines how much probability every candidate region had before truncation. A support interval by itself cannot specify a truncated law: different parent densities on the same interval yield different outputs.^[1]^[2]
Retained event. A rule identifies the values kept, such as \(a<X<b\) for a truncated normal or \(X>0\) for a zero-truncated Poisson. Declaring which endpoints count is essential for discrete mass at boundaries.^[1]^[3]
Positive retained mass. The parent probability \(P_X(A)\) is the normalizer and must be greater than zero. Without it, ordinary conditionalization would divide by zero; merely writing a bounded set does not construct a probability law.^[1]^[3]
Conditional reallocation. The new law has zero probability outside \(A\) and assigns \(P_X(B\cap A)/P_X(A)\) inside. Discarding the outside without this division leaves total mass below one whenever the discarded region had positive mass.^[1]^[2]
Target-sensitive interpretation. In empirical uses, an estimate about retained observations is not automatically an estimate about the unselected population. This is a common inferential consequence, not a constitutive part of the mathematical law; Stata's labor-hours example makes the distinction explicit.^[1]

What It Is Not¶

A truncated distribution is not any distribution with finite or one-sided support. A uniform law originally defined on \([0,1]\) is bounded, but it is not thereby a truncation of a specified wider parent law. To call it truncated, one must identify the parent, retained event, and resulting conditional normalization. Nor is truncation simply a plot cropped to show interesting values: a cropped display can leave the underlying law unchanged.

It is not censoring. If a measurement below a detection limit is still represented as “below the limit,” that unit contributes incomplete information; it has not vanished from the sampling mechanism. A truncated dataset instead selects units by range and models the law conditional on being included. Whether a particular study knows how many units were excluded depends on its data source and frame; ignorance of that count is not part of the distribution's definition.^[1]

It is also not just a truncated normal. The normal case retains a rescaled normal kernel, but a discrete Poisson law conditioned on positive counts has the same abstract construction with a mass function rather than a density. The live Truncated Normal Distribution and Zero-truncated Poisson Distribution nodes are narrower instances, not alternative names for this family-independent identity.^[1]^[3]

Scope of Application¶

The construction applies wherever a parent probability law and a positive-probability retained event are specified. It can be left-, right-, or two-sided, continuous or discrete; the statement of the event—not the word “cutoff” alone—decides the law.^[1]^[2]^[3]

Probability theory and simulation. Conditioning a specified law to a range produces a valid new law for drawing or reasoning about admissible values. SciPy's general continuous operation can truncate a transformed normal or a Rayleigh distribution; it is not tied to normality.^[2]
Selected-sample statistics. A minimum-eligibility threshold, positive-only observation, or interval-limited record can make the observed values follow a conditional law, assuming the parent model and selection mechanism are appropriate. Stata's labor-hours example compares a positive-hours selected group with the broader source sample.^[1]
Count models. If only positive counts can enter the analyzed set, a Poisson parent can become zero-truncated. The actuar documentation defines the resulting probability masses on \(1,2,\ldots\), normalized by the nonzero probability.^[3]

Selection on unobserved variables, informative missingness, or a mixture of censored and truncated mechanisms may require a richer model; simply writing a truncation denominator does not solve those identification problems.

Clarity¶

The name forces three questions that a vague “restricted data” description can conceal: Which original law? Which values are retained? What mass did that event have? Answering them separates a valid conditional distribution from a visual crop, a restricted-domain function that fails to integrate to one, and a censored observation model. The calculation is exact once those inputs are known. If they are not known, writing the truncated density does not magically identify them.

It also clarifies the estimand. A mean taken under \(P_T\) is the mean given inclusion. Stata notes that a selected-sample mean may be suitable when the question concerns selected workers' hours, yet misleading when the target is hours over working and nonworking women alike. The same number is not two different estimands merely because it appears in one dataset.^[1]

Manages Complexity¶

Potentially many excluded outcomes collapse into one retained event and its parent probability. The law's complete effect on any retained event \(B\) is \(P_X(B\cap A)/P_X(A)\). For a normal interval, the denominator reduces to a difference of cumulative-normal probabilities; for a positive Poisson count it reduces to \(1-e^{-\lambda}\). This compact normalizer ensures the reweighted densities or masses sum to one without separately recounting every excluded case.^[1]^[3]

That compression also tells analysts what information was lost. The conditional law describes relative probability inside the retained region; it does not, by itself, reveal the number of excluded units in an empirical study or prove that a particular parent family generated all candidates. If those questions matter, the sampling frame or a model for the unobserved portion must supply more than the truncated observations.

Abstract Reasoning¶

Given a candidate truncation claim, reason in order. First type the parent law and retained event. Then calculate the event's probability under the parent and verify it is positive. Finally compare the proposed output to the conditional law: outside mass must be zero and inside ratios preserved, because \(P_T(B_1)/P_T(B_2)=P_X(B_1)/P_X(B_2)\) for positive-probability subsets wholly within \(A\). This ratio invariant distinguishes conditioning from arbitrary reshaping of a bounded density.

The construction makes a diagnostic inference possible: if a positive-only count sample is fitted with an unmodified Poisson likelihood, the zero mass of the parent has been ignored in the observed-law normalization. A conditional likelihood must account for retention if the inferential target is the underlying parent parameter. Likewise, if thresholded units remain as boundary-coded records, a censoring likelihood, not a truncation likelihood, may be relevant. The data-generation facts decide which inference is justified.^[1]^[3]

Knowledge Transfer¶

Transfer is literal across probability-law families. The same conditioning-and-renormalizing rule applies to normal interval laws, positive Poisson counts, and other parent distributions when the retained event has positive mass. Only the parent kernel and normalizer change. SciPy's operation illustrates the continuous case beyond normality, while actuar's zero-truncated Poisson illustrates a discrete one.^[2]^[3]

The wider idea of applying a boundary and keeping the admissible part belongs to live Truncation. But outside probability theory, that operation need not conserve total probability by renormalization. Calling a cropped image, shortened string, or filtered list a “truncated distribution” is therefore analogy or a different typed use, not literal travel of this domain abstraction. The live Conditional Probability captures the event-level ratio; this entry constructs the entire resulting law.

Examples¶

Canonical: interval-truncated normal law¶

Let \(X\sim N(\mu,\sigma^2)\) and retain only \(a<X<b\), with nonzero interval mass. Stata gives the new density \(f_T(x)=f_N(x)/[\Phi((b-\mu)/\sigma)-\Phi((a-\mu)/\sigma)]\) inside the interval and zero outside. The normal's original parameters still define the kernel; the truncated law's mean need not equal \(\mu\). This is not merely a normal curve clipped at the ends—the denominator changes every retained density ordinate.^[1]

Mapped back: the parent probability law is \(N(\mu,\sigma^2)\); the retained event is \(a<X<b\); the positive retained mass is the normal-CDF difference; conditional reallocation divides the surviving normal density by that difference; and any target-sensitive interpretation distinguishes properties of the retained group from those of the original normal population.

Applied: zero-truncated Poisson counts¶

Let \(X\sim\mathrm{Poisson}(\lambda)\) with \(\lambda>0\), and retain only \(X>0\). Because \(P(X=0)=e^{-\lambda}\), the retained mass is \(1-e^{-\lambda}\). The resulting law has \(P_T(X=k)=\lambda^k/[k!(e^{\lambda}-1)]\) for integers \(k\ge1\), and zero at \(k=0\). This is precisely the zero-truncated Poisson family documented by the authors of the actuar package; it is a discrete instance of the same operation.^[3]

Mapped back: the parent probability law is Poisson on nonnegative integers; the retained event is \(X>0\); the positive retained mass is \(1-e^{-\lambda}\); conditional reallocation rescales all positive Poisson masses; and the target-sensitive interpretation is that an observed positive-only count sample has a conditional count law, not an ordinary Poisson law with zeros somehow forgotten.

Structural Tensions¶

T1: Selected-sample fit vs. whole-population recovery. Conditioning gives an honest law for observed eligible values. But its mean and other summaries can differ from those of the parent population, so using it to answer a whole-population question trades target fidelity for available-data convenience. A parent-law assumption or information about excluded units may be required to infer beyond the retained group.^[1] Diagnostic: Is the quantity sought conditional on inclusion, or does it describe all original candidates?

T2: Compact interval notation vs. exact boundary semantics. Writing \([a,b]\) makes a model easy to communicate, yet an atom at \(a\) or \(b\) can make open and closed endpoints produce different normalizers. Explicit event notation costs a few symbols but prevents a substantive discrete-law error. Diagnostic: Does the parent assign positive mass at a boundary, and is that value retained?

T3: Generic truncation vs. domain-specific probability law. The generic idea “cut off the outside” makes the family resemblance visible, but stopping there omits conditional normalization. Conversely, treating every bounded density as a truncation imports a parent law where none was specified. Diagnostic: Can the original law, retained event, and reweighting denominator all be exhibited?

Structural–Framed Character¶

Truncated Distribution is structural-leaning within its probabilistic domain. On evaluative weight it has none: conditioning neither approves nor condemns an outcome. On human-practice dependence, the equation defines a mathematical law even if no observer ever samples it; empirical truncation mechanisms are applications, not prerequisites. On institutional origin, the statistical vocabulary and tools are human-developed, but the conditional ratio follows from a coherent probability law rather than an agency's rule. Its mathematical identity is thus not a merely framed convention.^[1]^[2]

Its vocabulary travel is narrower than the very broad English word “truncation.” A cropped image or shortened sequence literally instantiates live Truncation, yet no probability mass is necessarily renormalized. On import versus recognition, normal and Poisson cases recognize the same conditional-law relation; crossing to nonprobabilistic boundaries usually imports an analogy instead. The portable skeleton of boundary-retain-discard belongs to Truncation, while coherent allocation of event probabilities belongs to Probability and the event ratio to Conditional Probability. The named entry adds law-level probability machinery and should not be promoted to a free-floating prime on the strength of a shared word. Its character: a source-grounded, evaluatively neutral mathematical construction with literal cross-family reach inside probability theory but a domain-specific normalization rule.

Structural Core vs. Domain Accent¶

Skeletal relation. A declared boundary retains one part of a larger possibility space and excludes another; live Truncation already represents that substrate-neutral operation. A second broad skeleton is conditional selection from a larger set of possibilities, represented in probability terms by live Conditional Probability.

Domain-bound mechanism. A truncated distribution must start from a probability law, compute a positive retained-event mass and divide the parent's event probabilities by that mass. The whole result remains a normalized law. Neither a cut string nor a cropped plot carries this density/pmf and measure-level requirement. Nor does ordinary conditional probability by itself name the output as a distribution over all retained outcomes.^[1]^[3]

Why this is not a prime. Literal reach across normal, Poisson, and other families is broad within one typed mathematical domain. The cross-domain language that survives removal of probability machinery—retaining an admissible part—is already the prime Truncation. The distinctive conditional-normalization mechanism does not run in domains without an underlying law and event mass. Calling the named object a prime would either duplicate the live prime or strip away the feature that identifies this node.

This entry is a kind of Probability Distribution. A truncated distribution is a probability law conditioned on a positive-probability retained event.

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

Not to Be Confused With¶

Censoring: the record remains but some value information is bounded, masked or limit-coded. The likelihood can include information that a value lies beyond the limit. Truncation instead describes the law conditional on a value entering the retained range.^[1]
Truncated normal distribution: specifically a normal parent law conditioned on a stated interval or half-line. This node is parent-family-independent; a Poisson example proves the distinction.^[1]^[3]
Zero-truncated Poisson distribution: the discrete special case that keeps positive integer counts and removes zero, rather than the generic operation across families.^[3]
Bounded-support distribution: a law can be natively bounded without being represented as conditioning a broader parent law.
Conditional probability: the ratio for one event given another, rather than the complete resulting law across all values.
Truncation prime: the more general boundary-retain-discard transformation, which does not demand a probability measure or normalization.

References¶

[1] StataCorp, Truncated regression [R] truncreg, current official manual, pp. 3–6, especially the conditional-normal density equation, Example 1, technical note and “Truncation and censoring are different concepts” paragraph. Original manual; the full pages and formulas were inspected. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u

[2] SciPy developers, scipy.stats.truncate, v1.15.2 API documentation, definition and worked examples. Original implementation documentation for continuous laws. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g

[3] Vincent Goulet and actuar package contributors, The Zero-Truncated Poisson Distribution, actuar reference manual, “Details” and “Examples.” The HTML's first exponential rendering is garbled; the equivalent second pmf and example establish the formula used here. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m