The r‐hypergeometric distribution: Characterization, mathematical methods, simulations, and applications in sciences and engineering
Bibliographic record
Abstract
In this article, we introduce a novel univariate discrete distribution called the r‐hypergeometric model. This distribution has the sampling characteristics with replacement of the binomial distribution and no‐order sampling of the hypergeometric distribution. Mathematical expressions are obtained for computing probabilities, the mode, and moments of the new distribution. Two simulation algorithms are proposed using the acceptance‐rejection and inverse‐transform methods and computational features to generate values of an r‐hypergeometric distributed random variable. Python and R codes are implemented to perform computational experiments. Applications of mathematical methods based on the new distribution in sciences and engineering employing simulated and real‐world data sets are provided. A comparison with existing distributions is also included. In addition to the mathematical results, some findings obtained from our study are related to a better computational performance of the inverse‐transform method. Also, we identify applications of our model that are not covered by the traditional count distributions. In addition, we establish distinct probabilities for the same event under the binomial, hypergeometric, and r‐hypergeometric distributions, with their means also being distinct, but their variances, skewness, and kurtosis converge to the same value.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.010 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".