Saddlepoint Approximation to Cumulative Distribution Function for Poisson–Exponential Distribution
Bibliographic record
Abstract
The random sum distribution is a key role in probability theory and its applications as well, these applications could be used in different sciences such as insurance system, biotechnology, allied health science, etc. The statistical significance in random sum distribution initiates when using the applications of probability theory in the real life, where the total quantity X can be only observed, which is included of an unknown random number X of random contributions. Saddlepoint approximation techniques overcome this problem. Saddlepoint approximations are effective tools in getting exact expressions for distribution functions that are not known in closed form. Saddlepoint approximations usually better than the other methods in which calculation costs, but not necessarily about accuracy. This paper introduces the saddlepoint approximations to the cumulative distribution function for random sum Poisson- Exponential distributions in continuous settings. We discuss approximations to random sum random variable with dependent components assuming existence of the moment generating function. A numerical example of continuous distributions from the Poisson- Exponential distribution is presented.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 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 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".