[harsh-thakkar7]

Resolves #15780.

Description

This pull request:

• fixes stats/base/dists/geometric/mgf to return p / (1 - (1-p)e^t), the MGF for the number of failures before the first success, which is the parameterization used by the rest of the geometric packages (pmf, mean, variance, etc.);

• previously it returned pe^t / (1 - (1-p)e^t), the MGF for the number of trials, so every value was off by a factor of e^t;

• applies the same fix to the C implementation in src/main.c, and updates the affected doctests in lib/index.js, lib/native.js, lib/factory.js, the README equation, and the README/C example values;

• regenerates the expected values in test/fixtures/julia/{small_p,large_p}.json, which had been computed with the old formula (the x and p inputs are unchanged);

• updates the p = 1 test expectation in test.mgf.js / test.native.js: X is degenerate at 0 when p = 1, so the MGF is 1, not e^t.

Related Issues

This pull request has the following related issues:

• [BUG]: stats/base/dists/geometric/mgf returns the MGF for the number of trials instead of the number of failures #15780
• fix: return MGF for number of failures in stats/base/dists/geometric/mgf #15781 (an earlier PR for this issue that has been inactive for about a month; happy to withdraw this one if that PR is being revived)

Questions

No.

Other

Verification with p = 0.3:

• mgf(0.1, 0.3) now returns 1.3252033149682356, matching the numerically integrated sum_k e^{0.1k} pmf(k, 0.3) from the issue (1.3252);
• the numeric derivative (mgf(1e-6, 0.3) - mgf(-1e-6, 0.3)) / 2e-6 is 2.3333, matching mean(0.3) (2.3333); before the fix it was 3.3333;
• mgf(0.5, 1.0) now returns 1 (X is always 0 when p = 1); before the fix it returned 1.6487;
• the domain check is unchanged (mgf(0.8, 0.5) still returns NaN).

Checklist

• Read, understood, and followed the contributing guidelines.

AI Assistance

• Yes

• No

• Code generation

• Documentation (including examples)

• Research and understanding

Disclosure

This PR was prepared with Qoder AI assistance under the contributor's direction. All changed code and examples were executed locally to verify the values above.

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Coverage Report

Package	Statements	Branches	Functions	Lines
stats/base/dists/geometric/mgf	$\\color{green}288/288$ $\\color{green}+100.00\\%$	$\\color{green}19/19$ $\\color{green}+100.00\\%$	$\\color{green}4/4$ $\\color{green}+100.00\\%$	$\\color{green}288/288$ $\\color{green}+100.00\\%$

The above coverage report was generated for the changes in this PR.