add: beta probability distribution

[Calluumm]

Adds a beta distribution to stats
This is a modernisation/continuation of #286 all credit to the original author.

Comparatively to #286 :
Tests are standardised and modernised to current stdlib standards
Beta calculations are moved to the gamma specialfunctions module
Documentation changed to modern standards

Build tests passed fine, calculations matched that of scipy

I think beta calculations are best suited in the gamma specialfunctions rather than their own file or within the beta distribution module for code efficiency and later use
@sebastian-mutz

[Calluumm]

locally wrote it relying on the changes made in #1133 will leave as draft for now

[codecov]

Codecov Report

❌ Patch coverage is 0% with 32 lines in your changes missing coverage. Please review.
✅ Project coverage is 68.68%. Comparing base (4c8521d) to head (f36788e).

Files with missing lines	Patch %	Lines
...ample/stats_distribution_beta/example_beta_rvs.f90	0.00%	14 Missing ⚠️
...ample/stats_distribution_beta/example_beta_cdf.f90	0.00%	9 Missing ⚠️
...ample/stats_distribution_beta/example_beta_pdf.f90	0.00%	9 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1139      +/-   ##
==========================================
- Coverage   68.81%   68.68%   -0.14%     
==========================================
  Files         408      411       +3     
  Lines       13726    13758      +32     
  Branches     1552     1552              
==========================================
+ Hits         9446     9450       +4     
- Misses       4280     4308      +28     

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[Copilot]

Pull request overview

Adds a Beta probability distribution implementation to stats, backed by new Beta-related special functions colocated in stdlib_specialfunctions_gamma, along with tests, examples, and specification docs.

Changes:

• Introduces stdlib_stats_distribution_beta with rvs_beta, pdf_beta, and cdf_beta (real + complex overloads).
• Adds beta, log_beta, and incomplete_beta to stdlib_specialfunctions_gamma, plus corresponding unit tests.
• Wires new module/tests/examples/docs into CMake and the specs index.

Reviewed changes

Copilot reviewed 13 out of 13 changed files in this pull request and generated 4 comments.

Show a summary per file

File	Description
src/stats/stdlib_stats_distribution_beta.fypp	New Beta distribution implementation (rvs/pdf/cdf).
src/specialfunctions/stdlib_specialfunctions_gamma.fypp	Adds Beta-related special functions used by the distribution.
test/stats/test_distribution_beta.fypp	New Beta distribution test coverage.
test/stats/CMakeLists.txt	Registers the new Beta distribution test.
test/specialfunctions/test_specialfunctions_gamma.fypp	Adds tests for beta, log_beta, and incomplete_beta.
src/stats/CMakeLists.txt	Includes the new Beta distribution module in the stats build.
example/stats_distribution_beta/example_beta_rvs.f90	Example usage for rvs_beta.
example/stats_distribution_beta/example_beta_pdf.f90	Example usage for pdf_beta.
example/stats_distribution_beta/example_beta_cdf.f90	Example usage for cdf_beta.
example/stats_distribution_beta/CMakeLists.txt	Builds the new Beta distribution examples.
example/CMakeLists.txt	Adds the Beta distribution examples subdirectory.
doc/specs/stdlib_stats_distribution_beta.md	New Beta distribution specification page.
doc/specs/index.md	Adds Beta distribution to the specs index.

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[sebastian-mutz]

Thanks, @Calluumm. Looks good and consistent with gamma and previous distribution modules.

• Since we moved away from normal statistical notation, I agree with the use of a and b instead of alpha and beta. (The former seems common; see Maltab and SciPy).
• I agree with Jose about the elemental declaration.
• Regarding the additions to the special functions gamma module: I support this implementation. Since beta is closely related to gamma, I think this works. I think the risk of creating confusion down the line is minimal, esp. if the person looking at the code has some understanding of these distributions.

[Calluumm]

This commit should address the above suggestions they can be pure elementals yes, locally passed everything

[jvdp1]

Thank you @Calluumm for this PR. The PR is taking a good shape. Please find below some suggestions.

[Calluumm]

I think I've resolved everything you suggested with this commit @jalvesz

@jvdp1 I made amends to the cdf example and subsequently the pdf example which had the same issues
ive pushed through the majority of the other suggestions too without problem I'm just stuck on the error handling changes to the beta distribution test, I've left comments on the suggestions directly

[Calluumm]

Applied all suggested changes with exception to using an error variable in tests. The other distribution functions don't so this would be standard to them.

If all of them move to use it it could be done in a seperate PR I think.