feat(Probability/Distributions/Exponential): add MGF, moments, and memoryless property

[JoaBjo]

feat(Probability/Distributions/Exponential): add MGF, moments, and memoryless property

Add the main analytic results for the exponential distribution:

• moment-generating function mgf id (expMeasure r) t = r / (r - t) for t < r
• mean ∫ x, x ∂(expMeasure r) = r⁻¹
• variance Var[id; expMeasure r] = r⁻¹ ^ 2
• ℒp membership for all p
• tail probability P(X > x) = exp (-(r * x))
• memoryless property P(X > s + t | X > s) = P(X > t)

The MGF is computed by reducing to the known improper integral ∫ exp(c * x) on Ioi,
and integrability is deduced by contradiction from the positive closed-form value.
The mean and variance are computed via the Gamma function integral
∫₀^∞ x^(n-1) exp(-r x) dx = Γ(n) / rⁿ. The memoryless property follows from
the exponential identity exp(-(r(s+t))) = exp(-rt) * exp(-rs).

---

[github-actions]

Welcome new contributor!

Thank you for contributing to Mathlib! If you haven't done so already, please review our contribution guidelines, as well as the style guide and naming conventions.

We use a review queue to manage reviews. If your PR does not appear there, it is probably because it is not successfully building (i.e., it doesn't have a green checkmark), has the awaiting-author tag, or another reason described in the Lifecycle of a PR.

If you haven't already done so, please come to https://leanprover.zulipchat.com/, introduce yourself, and mention your new PR.

Thank you again for joining our community.

[github-actions]

PR summary cde919c85c

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Probability.Distributions.Exponential	2584	2617	+33 (+1.28%)

Import changes for all files

Files	Import difference
Mathlib.Probability.Distributions.Exponential	33

---

Declarations diff

+ Iio_subset_integrableExpSet_id_expMeasure
+ integrable_exp_mul_expMeasure
+ integral_expMeasure_eq_integral_density
+ integral_exp_mul_expMeasure
+ integral_id_expMeasure
+ integral_sq_expMeasure
+ measure_Ioi_expMeasure
+ memLp_id_expMeasure
+ memoryless_expMeasure
+ mgf_id_expMeasure
+ variance_id_expMeasure
+ variance_integral_expMeasure
+ zero_mem_interior_integrableExpSet_id_expMeasure

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for scripts/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[Timeroot]

These show ... by rfl can be exchanged for [Set.Ioi.eq_1], which I think would be cleaner!

[JoaBjo]

I might be misunderstanding you, but I don't think that works