[DOC] add explicit formulae to _formula_docs for Logistic and Weibull distributions

[itsnevu]

Reference Issues/PRs

Towards #1120.

Does not overlap with #1123 (Uniform, @Vidit-lab) or #968 (Pareto). Claimed on the issue before starting.

What does this implement/fix? Explain your changes.

Populates _formula_docs for two distributions that previously had none:

• Logistic (skpro/distributions/logistic.py)
• Weibull (skpro/distributions/weibull.py)

Each dict covers all seven applicable keys: pdf, log_pdf, cdf, ppf, mean, var, energy, following the Normal example and the extension template.

I picked these two because they are native BaseDistribution implementations with elementary closed-form ppf and exact mean/var, so every key is fillable without guessing. I looked at Poisson, Binomial and Beta first and skipped them: they are _ScipyAdapter wrappers whose ppf has no closed form.

This PR also has to touch skpro/distributions/tests/test_docstring_injection.py. That test used Weibull as the UNHOOKED_CLASSES control group, so adding formulae to Weibull made test_unhooked_clean_fallback fail on its ".. math::" not in doc assertion. Weibull moves into HOOKED_CLASSES alongside Logistic, and Gamma becomes the new control - it has no _formula_docs and is unclaimed on #1120 at the time of writing. Happy to use a different control if you would prefer one less likely to be claimed.

References for the formulae

No AI-generated formulae.

Core pdf/log_pdf/cdf/ppf/mean/var are standard results from Johnson, Kotz & Balakrishnan, Continuous Univariate Distributions, 2nd ed. - Vol. 2 Ch. 23 (Logistic) and Vol. 1 Ch. 21 (Weibull); cross-checked against the NIST/SEMATECH e-Handbook 1.3.6.6.

The two energy formulae, stated explicitly since they need more than a citation:

• Logistic, E|X-Y| = 2s - already stated and derived in this repo, in the existing Logistic._energy_self docstring (logistic.py L223-233).
• Weibull, E|X-Y| = 2*lambda*Gamma(1 + 1/k)*(1 - 2**(-1/k)) - derived from the fact that for iid X, Y ~ Weibull(lambda, k), min(X, Y) ~ Weibull(lambda * 2**(-1/k), k), so E|X-Y| = 2*(E[X] - E[min]).

Both reproduce the Analytic column already tabulated in this repo's own skpro/distributions/energy_formulae.md - Logistic(0,1) = 2.000000 and Weibull(1,2) = 0.519140 - which I treat as independent confirmation rather than my own arithmetic checking itself.

Does your contribution introduce a new dependency? If yes, which one?

No.

What should a reviewer concentrate their feedback on?

• The choice of Gamma as the replacement control group in test_docstring_injection.py.
• Whether the energy key should be included at all, or left out until there is a dedicated review of energy formulae. Removing it is a clean deletion of one key per distribution; the other six are independent.
• Parameterisation, which I matched to the code rather than to any single textbook convention: Logistic.scale is the true scale s (so the sd is s*pi/sqrt(3)), and Weibull.scale is the multiplicative scale lambda, matching scipy.stats.weibull_min(c=k, scale=lambda). Neither is a rate. Weibull's documented support is x >= 0 because _pdf and _cdf both multiply by (x >= 0).

Did you add any tests for the change?

No new tests. The existing test_docstring_injection.py already covers exactly this change: test_hooked_math_injection now asserts that Logistic and Weibull render .. math::, and test_placeholder_removal_universal asserts no {formula_doc} leak. 95 tests pass locally.

Separately from the test suite, I verified every formula numerically against each class's own _pdf/_log_pdf/_cdf/_ppf/_mean/_var over several parameter settings (max deviation ~1e-15), and the energy closed forms against quadrature of 2 * integral F(t)(1-F(t)) dt.

Any other comments?

Two docstring errors in logistic.py that I did not touch here, since they are bugs rather than missing formulae, and I would rather not smuggle behaviour-adjacent corrections into a docs PR:

1. The class docstring at L17 gives the cdf as F(x) = 1/(1 + exp((x - mu)/s)) - the minus sign is missing, so that is the survival function, not the cdf. The code is correct.
2. _var at L79 writes \pi^3 where the code correctly computes pi**2.

Happy to fold both into this PR if you would prefer, or I will open a separate [BUG] issue.

On LogNormal, which I also claimed on the issue: lognormal.py is the only one of the three files with CRLF line endings, and editing it through the GitHub web editor normalises them to LF, turning a ~60 line addition into a 265-line whitespace diff. Rather than mix a line-ending change into a docs PR, I am leaving it for a follow-up done as a local commit. Worth noting that the CRLF endings in that one file may be unintentional in an otherwise LF repo.