fix: return MGF for number of failures in stats/base/dists/geometric/mgf

[SajalDevX]

Resolves #15780.

Description

What is the purpose of this pull request?

This pull request:

• fixes stats/base/dists/geometric/mgf (JS and C) so that it returns the MGF of the number of failures before the first success, p / (1 - (1-p) e^t). That is the convention used by the other geometric packages, such as pmf, mean, and variance. Before this fix it returned p e^t / (1 - (1-p) e^t), the MGF of the number of trials, so every value was off by a factor of e^t. For example, mgf( 0.5, 1.0 ) returned e^0.5 instead of 1, and M'(0) was 1/p instead of mean( p ) = (1-p)/p.
• regenerates the expected values in test/fixtures/julia/{small,large}_p.json, keeping the same x and p inputs, and changes the p = 1 test to expect 1.
• updates the documented equation and the example outputs in geometric/mgf and geometric/ctor.

Related Issues

Does this pull request have any related issues?

This pull request has the following related issues:

• #{{TODO: add related issue number}}

Questions

Any questions for reviewers of this pull request?

The original fixtures were generated with Distributions.jl 0.23.8 (per REQUIRE). That version used the same trials-based formula, which is why the old tests passed. I didn't have Julia available, so I regenerated expected with 60-digit decimal arithmetic for p / (1 - (1-p) e^t), rounded to the nearest double. Current Distributions.jl computes the same quantity as p / (p - (1-p) expm1(t)). Should I bump REQUIRE to a newer Distributions.jl version, or would you rather regenerate the fixtures with runner.jl?

Other

Any other information relevant to this pull request? This may include screenshots, references, and/or implementation notes.

• Testing: I ran test.js, test.mgf.js, and test.factory.js for geometric/mgf, plus geometric/ctor/test/test.js. With the new fixtures and the old code, 2000 fixture assertions failed in each of test.mgf.js and test.factory.js. After the fix, everything passes (2016/2016 and 2014/2014; ctor 67/67). I couldn't build the native add-on, so test.native.js was skipped. To check the C change, I compiled src/main.c on its own with libm stand-ins for exp, ln, and is_nan. It gives the same results as the JS version: 1.2843610871738946 for (0.2, 0.5) and 1 for (0.5, 1.0).
• The equation SVG in mgf/docs/img still shows the old formula. I only changed the raw equation text in the README, so the SVG will need to be regenerated with the usual tooling.

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Used Claude to help analyze the code and draft the fix; I reviewed and tested the change myself.

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[kgryte]

I commented on the original issue: #15780 (comment)

We first need to determine whether it is actually a bug and whether we want to keep the current formulation.

[Planeshifter]

We should bump REQUIRE to a newer Distributions.jl version here and regenerate the fixtures from that. Would you like to give that a shot, @SajalDevX, or shall we take this PR over? Thanks for your contribution!

[SajalDevX]

Sure, done! I bumped REQUIRE to Distributions 0.25.131 / julia 1.10.10 / JSON 1.10.0 and regenerated large_p.json and small_p.json with runner.jl.