[DOC] Add formula docs for Uniform distribution

[Vidit-lab]

Reference Issues/PRs

Part of #1120

What does this implement/fix? Explain your changes.

Populates the _formula_docs dictionary for Uniform (skpro/distributions/uniform.py) covering all seven exact methods (pdf, log_pdf, cdf, ppf, mean, var, energy).

Refactors standalone formula attributes (_pdf_formula_doc, _log_pdf_formula_doc, and _cdf_formula_doc) into the single _formula_docs structure.

Mathematical References & Derivations

Since the closed-form math for _pdf, _log_pdf, _cdf, _ppf, _mean, _var, _energy_self, and _energy_x is already implemented in uniform.py, the formulas in _formula_docs document these existing operations:

1. Standard Properties (pdf, log_pdf, cdf, ppf, mean, var):

   • Reference: Bertsekas, D. P., & Tsitsiklis, J. N. (2008). Introduction to Probability (2nd ed.). Athena Scientific, Chapter 3 (Continuous Random Variables).

2. Self-Energy ($\mathbb{E}[|X - Y|]$):

   For $X, Y \stackrel{\text{i.i.d.}}{\sim} U(a,b)$ with joint density $f(x, y) = \frac{1}{(b - a)^2}$:

   $$\mathbb{E}[|X - Y|] = \frac{1}{(b - a)^2} \int_{a}^{b} \int_{a}^{b} |x - y| , dx , dy$$

   Exploiting symmetry across $x = y$:

   $$\mathbb{E}[|X - Y|] = \frac{2}{(b - a)^2} \int_{a}^{b} \left( \int_{a}^{x} (x - y) , dy \right) dx = \frac{2}{(b - a)^2} \int_{a}^{b} \frac{(x - a)^2}{2} , dx = \frac{b - a}{3}$$

   Matches _energy_self return value (upper - lower) / 3.

3. Point Energy ($\mathbb{E}[|X - x|]$):

   Evaluated for constant $x$:

   $$\mathbb{E}[|X - x|] = \int_{a}^{b} |t - x| \frac{1}{b - a} , dt$$

   For $x \in [a, b]$, splitting the integral at $t = x$ yields:

   $$\mathbb{E}[|X - x|] = \frac{(b - x)^2 + (a - x)^2}{2(b - a)}$$

   For $x < a$ or $x > b$, direct evaluation yields $\left| x - \frac{a + b}{2} \right|$.

   Matches _energy_x logic.

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

No.

What should a reviewer concentrate their feedback on?

• LaTeX rendering formatting inside _formula_docs.

Did you add any tests for the change?

Added Uniform to HOOKED_CLASSES in test_docstring_injection.py.

Any other comments?

None.

PR checklist

• I've added myself to the list of contributors in the skpro root directory.
• The PR title starts with [DOC].