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Inference practice and worked checks

Attempt each prompt before reading its calibration below. Preserve the model, algebra and changed-case explanation, not only a number.

Prompts​

  1. Let X be uniform on [0,1] and Y=X³. Derive the full CDF, density on its support and mean. Check the density by an improper integral.
  2. Let f(x,y)=2 on 0<y<x<1. Derive both marginals, E[Y|X=x] and E[Y]. Explain dependence from support.
  3. For uncensored independent exponential durations [1,1,2,4], derive the rate MLE. Convert the units from seconds to milliseconds. Explain the all-zero case and why dropping timeouts changes the likelihood.
  4. With a Beta(2,3) prior and three successes in six conditionally independent trials, derive posterior parameters, posterior mean and next-trial predictive probability. Show an update in batches of (1 success,2 trials) and (2 successes,4 trials) gives the same answer.
  5. Under independent fair within-pair treatment assignment and the sharp no-effect null, enumerate the absolute-sum randomization p-value for [1,1,2]. Explain ties, resolution, and why it is not a posterior probability.
  6. For a uniform prior and zero successes in two trials, derive and integrate the posterior density. Find a one-sided 95% credible upper bound. Compare its interpretation with a confidence limit without claiming the numerical values must match.

Calibration after your attempt​

  1. CDF is 0 below 0, y^(1/3) on [0,1], and 1 above 1. Density is (1/3)y^(−2/3) on (0,1), whose integral is 1. E[Y]=∫₀¹x³dx=1/4, not (1/2)³.
  2. f_X(x)=2x, f_Y(y)=2(1−y), on (0,1). Conditional density of Y given x is 1/x on (0,x), so conditional mean x/2 and unconditional mean ∫₀¹(x/2)2x dx=1/3. The product of marginals would allow impossible pairs.
  3. Rate 4/8=1/2 per second, 1/2000 per millisecond. All-zero data give no finite maximizer in λ>0. Censored times contribute survival probabilities; the simple complete-observation product is insufficient.
  4. Posterior Beta(5,6), mean and next-trial success probability 5/11. Sequential parameters (3,4) then (5,6). Updating twice with the same batch falsely counts extra evidence.
  5. Absolute sums across eight assignments are 4,0,2,2,2,2,0,4 (order irrelevant). Two reach the observed 4, so p=1/4. Repeated statistic values retain their assignment multiplicities.
  6. Beta(1,3) posterior has density 3(1−p)² and CDF 1−(1−p)³. Upper bound 1−0.05^(1/3), approximately 0.632. It is conditional on the uniform prior and the Bernoulli sampling model.