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Independent Module Assessment

Attempt after the practice and labs. Allow90 minutes; a calculator is permitted. Close worked examples, reference code, and answer keys. Write assumptions and derivations, not just numbers. Record assistance and interruptions. This is a local competency check, not a validated percentile ranking.

Each item is worth4 points: one for model/assumptions, two for derivation, one for interpretation or limitation. See assessment rules before starting; open solutions afterward.

  1. Sampling protocol. Two independent fair coins are flipped. A program selects one position uniformly and reveals H. Find the probability both flips were H. Contrast this with conditioning on "at least one H." (PS1)
  2. Base rates. A detector sees prevalence=.04,sensitivity=.75,false-positive rate=.02. For10,000 requests, construct the expected count table and compute precision. Explain why precision is not sensitivity. (PS2)
  3. Stopping. An operation independently fails with probability .1 on each attempt. At most three total attempts are made, stopping on success. Derive expected attempts, failures, backoffs, and eventual success. Explain why a binomial model is wrong. (PS3)
  4. Finite sampling. Uniformly choose two of five nodes, two of which are faulty. Derive the faulty-count PMF, mean, and variance. Compare to a with-replacement experiment. (PS4)
  5. Dependence. X,Y have variance4 each and covariance3. Find Var((X+Y)/2). Compare with the independent case; explain why averaging does not always halve variance. (PS5)
  6. Continuous models. X is uniform on[0,4]. Give its density, P(1<=X<=3), its mean and variance, and its lower99th quantile. Explain why P(X=2)=0. (PS6)
  7. Inference. For i.i.d. observations with finite mean and positive finite variance, distinguish LLN, Var(mean), and CLT. Explain why k=0 failures in20 trials is not proof p=0, and what repeated-trial confidence coverage means. (PS7)
  8. Engineering judgment. Averages for small/large requests are A=10/100 ms,B=12/110 ms. A sees10% small requests, B sees90%. Compute raw and50/50 standardized means. Propose a sampling unit and fixed stopping rule for a new comparison, and state two missing facts that block a causal conclusion. (PS8)

Changed-case recheck​

After3–7 days, work two previously weak competencies without the key. The reviewer chooses new values and one changed assumption. Examples: change maximum attempts to four; change sampling to replacement; introduce a shared outage; change randomization from requests to sessions. Before computing, identify which earlier formulas still apply. Use transfer prompts if studying alone, labeling the result self-assessed.