On the Convergence of Hypergeometric to Binomial Distributions
Bibliographic record
Abstract
This study presents a measure-theoretic approach to estimate the upper bound on the total variation of the difference between hypergeometric and binomial distributions using the Kullback-Leibler information divergence. The binomial distribution can be used to find the probabilities associated with the binomial experiments. But if the sample size is large relative to the population size, the experiment may not be binomial, and a binomial distribution is not a good choice to find the probabilities associated with the experiment. The hypergeometric probability distribution is the appropriate probability model to be used when the sample size is large compared to the population size. An upper bound for the total variation in the distance between the hypergeometric and binomial distributions is derived using only the sample and population sizes. This upper bound is used to demonstrate how the hypergeometric distribution uniformly converges to the binomial distribution when the population size increases relative to the sample size.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.033 | 0.161 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.007 | 0.003 |
| Science and technology studies | 0.002 | 0.011 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".