Bernoulli sampling¶
Select each eligible unit independently with the same preset inclusion probability, allowing the sample size to vary.
Core Idea¶
Bernoulli sampling gives each eligible unit the same preset probability \(p\) of entering a sample, and makes the inclusion decisions independently.[^ref-73510208b8f7] It does not fix the sample size. For a finite frame of \(N\) units under the pure design, the selected count has a binomial distribution with expected value \(Np\).[^ref-73510208b8f7] The draw strictly presupposes a Probability Measure over the possible patterns of inclusion; the draw is a procedure, not itself a measure.
Scope of Application¶
A survey can use the rule within a defined stratum of business units, while a database can apply it to individual rows. Statistics Canada's Integrated Business Statistics Program uses Bernoulli selection as a default component but also uses strata and a fixed-size fallback when a selected count is too small.[^ref-45dcb60d8b09] PostgreSQL's TABLESAMPLE BERNOULLI makes independent row decisions; its different SYSTEM method selects storage blocks.[^ref-3deafd196aa9] The whole Statistics Canada program is therefore not one global common-\(p\) Bernoulli draw.
Clarity¶
The Bernoulli trial determines whether a unit enters the sample. It is not a claim that the unit's survey answer is binary. Equal inclusion chance alone is insufficient: a fixed-size simple random sample can give equal marginal chances while coupling the unit decisions.[^ref-73510208b8f7] Bernoulli sampling can return more or fewer than its expected count, including an empty sample. It also does not guarantee that the realized subset represents every feature of a target population.
Manages Complexity¶
Specify the eligible units and one \(p\), then make an independent include-or-exclude decision for each. The chance of a particular selected set depends on how many units it contains: \(p^{n_s}(1-p)^{N-n_s}\) for a set of size \(n_s\).[^ref-73510208b8f7] This short rule is enough to calculate the pure design's count distribution. It must be revised if an agency replaces a too-small draw with a different sampling method.[^ref-45dcb60d8b09] It also must be revised if a database samples blocks instead of individual rows.[^ref-3deafd196aa9]
Abstract Reasoning¶
The expected count \(Np\) is a planning value, not a promise about the result.[^ref-73510208b8f7] Before using the draw, one can calculate the risk of too few units and decide whether a fallback is needed. After a draw, one must identify the selection operation that actually produced the final units. Population inference also needs a separate account of frame coverage, weights and the target quantity. Selection is a related catalog pattern, but its full differential-retention signature does not give an all-instance parent edge for equal-\(p\) Bernoulli draws.
Knowledge Transfer¶
In a business survey, the units can be firms in a stratum.[^ref-45dcb60d8b09] In PostgreSQL, they are table rows.[^ref-3deafd196aa9] In each case, common \(p\) and independent unit-level decisions produce a variable-size selected set. This is a literal transfer of the statistical sampling rule. A block sample or a fixed-size fallback changes the joint inclusion law. The needed Probability Measure is the normalized product law on possible inclusion outcomes; the broader Probability and Measure concepts are inherited through that live catalog parent.
[^ref-73510208b8f7]: Ismo Teikari, Poisson Mixture Sampling in Controlling the Distribution of Response Burden in Longitudinal and Cross Section Business Surveys, Statistics Finland Research Reports 232 (2001), printed pp. 23–26, PDF pp. 24–27. https://otos.stat.fi/bitstreams/950353df-68a8-4fd6-bda6-05b207e094f5/download . Full Statistics Finland PDF inspected; Bernoulli membership indicators, binomial count and equal-probability relation to Poisson sampling are stated there.
[^ref-45dcb60d8b09]: Statistics Canada, “An Overview of the Integrated Business Statistics Program Sampling Methodology,” especially “Sample selection and coordination,” “Allocation,” and “Must-Take Units.” https://www.statcan.gc.ca/en/statistical-programs/document/IBSP_D2_T9_V1 . Full official page inspected; the program includes stratification and non-Bernoulli components.
[^ref-3deafd196aa9]: PostgreSQL Global Development Group, PostgreSQL 10 Documentation, “SELECT,” FROM clause TABLESAMPLE paragraphs. https://www.postgresql.org/docs/10/sql-select.html . Full official documentation inspected; BERNOULLI operates on individual rows and SYSTEM on blocks.
Relationships to Other Abstractions¶
Current abstraction Bernoulli sampling Domain-specific
Parents (1) — more general patterns this builds on
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Bernoulli sampling presupposes Probability measure Domain-specific
Bernoulli sampling requires a normalized product probability measure over unit-inclusion outcomes.
Hierarchy path (1) — routes to 1 parentless root
- Bernoulli sampling → Probability measure → Probability → Measure → Set and Membership
Neighborhood in Abstraction Space¶
Bernoulli sampling sits in a sparse region of the domain-specific corpus (97th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2721 abstractions)
Nearest neighbors
- Truncated Regression Model — 0.77
- Binomial Proportion Estimation — 0.76
- Negative Hypergeometric Distribution — 0.76
- Systematic sampling — 0.76
- Multistage sampling — 0.75
Computed from structural-signature embeddings · 2026-10-17