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Expectation and Statistical Reasoning Practice

Allow 60–90 minutes. Every inference answer must distinguish an algebraic result, a modeling assumption, and an empirical observation. Use the separate key after your attempt.

  1. C1 — indicators. Four labeled requests independently choose among three servers uniformly. Define an indicator for each empty server. Derive the expected number of empty servers without claiming those indicators are independent.
  2. C2 — copied evidence. X is Bernoulli(.5). Compare the variance of the mean of100 independent draws with a100-row file repeating a single X. Show covariance terms or simplify the copied mean directly.
  3. C3 — theorem choice. For i.i.d. Exp(1) draws, state mean and variance of a batch average of size n. Identify which facts hold exactly for all n and what the CLT adds. Would n=30 guarantee an accurate Normal approximation for every finite-variance population?
  4. C4 — misleading interval. At n=1,p=.5, calculate actual coverage of the Wald interval. Then explain why an always-[0,1] interval has excellent coverage but little usefulness.
  5. C5 — workload. Using the lab fixture, compute raw means and standardized means for80% small requests. Can you calculate a confidence interval for the difference from this aggregate table? Can you infer a causal version effect? Explain what is missing.
  6. C6 — simulation precision. A coverage experiment performs5,000 independent reruns; its true coverage is .95. Compute the standard deviation of its measured coverage proportion. Is a result of .947 proof that the implementation is broken?

Repair an error by returning to its missing assumption, not just repeating arithmetic. For C2, additional independent sessions can add information; duplicate rows cannot. For C5, make the next measurement plan concrete using the experiment-design lesson.