Independent inference gates

These gates extend PS1–PS8; they do not replace them. Attempt without the reviewer key, source solutions or generated code. Score each gate 0–4: one point each for model/support, derivation or implementation, independent check, and changed-case/limitation explanation. Require at least 3/4 on every gate including the model/support point, matching the original module standard. Repair and recheck after 3–7 days; reuse existing retention sessions.

• PS9 — Continuous models: for X uniform on [0,1], derive the CDF/density of X⁴ and its mean. For joint density 2 on 0<x<y<1, derive f_X and E[Y|X=x]. Identify where the conditional expression is defined and why independence fails.
• PS10 — Likelihood: derive the exponential rate MLE for [2,3,5,6], show it is a maximum, state units, handle all-zero observations, and explain why censoring changes the model.
• PS11 — Bayesian parameter inference: update Beta(3,2) with two successes in seven trials. Give posterior parameters and next-trial probability. For a separate uniform-prior model with zero successes in three trials, derive a 95% one-sided credible upper bound. Distinguish that statement from frequentist coverage.
• PS12 — Randomization and reproducibility: for differences [2,2,3,0], enumerate the absolute-sum p-value under the stated within-pair randomized design and sharp null. Give the mean difference and interpret both without claiming equivalence or posterior probability. Pass the new lab’s checks, retain two seeded reports, and reproduce one function and a changed case unaided.

Completion also requires the extension memo with all requested assumptions and limitations. An exact computation under the wrong assignment design fails PS12. A passing reference implementation is setup evidence only. Record assistance, first attempt, repair, reviewer status and delayed recheck for each gate. Self-assessment remains explicitly unreviewed.