Distributions of ratios of random variables from the power-quadratic exponential family and applications
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Bibliographic record
Abstract
Abstract We derive the closed form exact expression of the density of the ratio X 1/X 2, where X 1 and X 2 are independent and belong to the very versatile family of power-quadratic exponential distributions. The case of X 2 as a gamma variable is studied in detail, and new applications found in different domains show the varied uses of the results presented. Keywords: Power-quadratic exponentialGammaRatio of random variablesHermite functionTricomi functionRayleighBeta type 2Student-type Acknowledgement This research was partially supported by NSERC Grant A9249 (Canada). The authors wish to thank Professor Jeannette Dolorese LeBlanc for very helpful discussions on this topic and an anonymous referee for suggesting an interesting general result, which is presented as theorem, as well as offering other useful comments.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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 it