[BUG] Incorrect variance computation and misleading capability tags in Hurdle

[ANANYA542]

Describe the bug
The Hurdle distribution contains two independent but related issues:

1. Incorrect variance formula in _var The current implementation computes: p * var_positive + p * (1 - p) * mean_positive However, the correct variance for a hurdle distribution is:
   $$
   \mathrm{Var}(Y) = p\sigma^2 + p(1-p)\mu^2
   $$
   The mean term is not squared in the implementation, violating the law of total variance and leading to severely underestimated variance.

2. Incorrect capability tagging Hurdle determines whether mean and var are exact by inspecting the base distribution (e.g., Normal). However, internally it wraps this distribution in LeftTruncated, which:
   does not implement exact mean/var instead relies on numerical integration (PPF-based approximation)
   As a result: Hurdle advertises exact capabilities but actually produces approximate results and triggers runtime warnings This leads to misleading API behavior and inconsistent capability metadata.

To Reproduce

import numpy as np
from skpro.distributions.normal import Normal
from skpro.distributions.hurdle import Hurdle

base = Normal(mu=10.0, sigma=1.0)
hurdle = Hurdle(p=0.5, distribution=base)

np.random.seed(42)
samples = hurdle.sample(100_000)
empirical_var = float(samples.var().iloc[0])

skpro_var = float(hurdle.var().iloc[0, 0])

print("Empirical variance:", empirical_var)
print("skpro variance:", skpro_var)

screenshot of the issue is attached below:

Expected behavior
Variance should follow:

$$ \mathrm{Var}(Y) = p\sigma^2 + p(1-p)\mu^2 $$

Empirical and analytical variance should match within numerical tolerance Hurdle should correctly advertise mean and var as approximate, not exact

Environment
OS: macOS
Python: 3.x
skpro: latest (main branch)
NumPy / Pandas: standard versions

Additional context
The discrepancy is structural, not due to numerical approximation The related ZeroInflated distribution correctly implements this variance formula, indicating inconsistency
The issue impacts: uncertainty estimation ,probabilistic metrics ,downstream model evaluation

Proposed fix
Correct variance formula:
return ( self.p * var_positive+ (mean_positive ** 2) * self.p * (1.0 - self.p))
Fix capability tagging. Inspect LeftTruncated(distribution) instead of the raw base distribution Downgrade mean and var to approximate capabilities

[ANANYA542]

@fkiraly — I think there’s an issue in Hurdle._var. The mean term isn’t squared, so it’s using p(1-p)μ instead of p(1-p)μ², which ends up underestimating the variance quite a bit (I’m seeing ~3 instead of ~25.5 in a simple example). Also noticed that it marks mean/var as exact based on the base distribution, but since it wraps it in LeftTruncated, those are actually computed via approximation and throw warnings.Seems like squaring the mean term and checking capabilities on the truncated distribution should fix it — happy to open a PR if that makes sense.