Mathematical statistics: derivation, decisions and model checks
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 180 minutes, as an unvalidated planning estimate. This collection contains 10 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Choose methods before opening guidance. Write the domain, assumptions and intermediate reasoning; a decimal answer or theorem name alone is insufficient.
These public mixed tasks require human mathematical review. Saved work and automated checks do not certify an independent examination, proof validity or semester-course equivalence.
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0 of 10 tasks have recorded responses. No overall mastery score is assigned.
Let X₁,…,Xₙ be iid N(θ²,1), θ∈R. Is θ identifiable? Reparameterize the identifiable quantity, derive its constrained MLE and prove consistency. Explain what can and cannot be concluded about a unique MLE of θ.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For iid exponential observations with rate λ and n>2, S=ΣXᵢ has Gamma(n,rate λ) density. Derive E[1/S] and E[1/S²], then compare squared-error risk of n/S and (n−1)/S. Give the comparison at n=5.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For iid Poisson(λ) observations with n≥2, construct the Rao–Blackwell improvement of 1{X₁=0} using S=ΣXᵢ. Prove sufficiency and completeness of S and identify the resulting UMVU target.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For iid N(θ,4), derive information and the Cramér–Rao bound for θ. For g(θ)=θ², analyze T=Xbar²−4/n: show unbiasedness, compute its variance and compare with the transformed bound. Explain the special behavior at θ=0.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For exponential rate λ, compare an asymptotic log-MLE interval with an exact interval based on S=ΣXᵢ. State the regularity reasoning for the asymptotic limit and give both interval formulas when n=8,S=4, leaving distribution quantiles symbolic.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For X∼Binomial(4,p), construct a most powerful level 1/16 test of p=1/4 against p=3/4. Allow randomization, compute its power exactly and explain whether it is also UMP against p>1/4.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For n≥2 iid normal observations with unknown positive variance, derive the likelihood-ratio statistic for H₀:μ=μ₀ and relate it to the two-sided t test and its inverted confidence interval. Why is a realized p-value not the power of the test?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
With prior Beta(2,3), observe four successes in six Bernoulli trials. Derive the posterior and its squared-error Bayes action. Then use loss 3(θ−a) when a<θ and (a−θ) when a≥θ: characterize the Bayes action and explain why it is not generally the posterior mean.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In a normal linear model with intercept and predictor x=(−5,−3,−1,1,3,5), derive the coefficient estimates, residual degrees of freedom and exact slope test. Explain independence of fitted and residual components and the relation between the t and partial F tests.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For the preceding six-point regression design, compute the largest leverage. Explain why an ordinary residual bootstrap can fail with heteroscedastic errors and distinguish a pairs or wild bootstrap. Finally consider iid Xᵢ=η+Zᵢ with unknown lower endpoint η and Zᵢ exponential with known positive rate λ. Derive the distribution of n(m−η), where m is the sample minimum, and diagnose the ordinary bootstrap approximation to this endpoint error.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Let X₁,…,Xₙ be iid N(θ²,1), θ∈R. Is θ identifiable? Reparameterize the identifiable quantity, derive its constrained MLE and prove consistency. Explain what can and cannot be concluded about a unique MLE of θ.
For iid exponential observations with rate λ and n>2, S=ΣXᵢ has Gamma(n,rate λ) density. Derive E[1/S] and E[1/S²], then compare squared-error risk of n/S and (n−1)/S. Give the comparison at n=5.
For iid Poisson(λ) observations with n≥2, construct the Rao–Blackwell improvement of 1{X₁=0} using S=ΣXᵢ. Prove sufficiency and completeness of S and identify the resulting UMVU target.
For iid N(θ,4), derive information and the Cramér–Rao bound for θ. For g(θ)=θ², analyze T=Xbar²−4/n: show unbiasedness, compute its variance and compare with the transformed bound. Explain the special behavior at θ=0.
For exponential rate λ, compare an asymptotic log-MLE interval with an exact interval based on S=ΣXᵢ. State the regularity reasoning for the asymptotic limit and give both interval formulas when n=8,S=4, leaving distribution quantiles symbolic.
For X∼Binomial(4,p), construct a most powerful level 1/16 test of p=1/4 against p=3/4. Allow randomization, compute its power exactly and explain whether it is also UMP against p>1/4.
For n≥2 iid normal observations with unknown positive variance, derive the likelihood-ratio statistic for H₀:μ=μ₀ and relate it to the two-sided t test and its inverted confidence interval. Why is a realized p-value not the power of the test?
With prior Beta(2,3), observe four successes in six Bernoulli trials. Derive the posterior and its squared-error Bayes action. Then use loss 3(θ−a) when a<θ and (a−θ) when a≥θ: characterize the Bayes action and explain why it is not generally the posterior mean.
In a normal linear model with intercept and predictor x=(−5,−3,−1,1,3,5), derive the coefficient estimates, residual degrees of freedom and exact slope test. Explain independence of fitted and residual components and the relation between the t and partial F tests.
For the preceding six-point regression design, compute the largest leverage. Explain why an ordinary residual bootstrap can fail with heteroscedastic errors and distinguish a pairs or wild bootstrap. Finally consider iid Xᵢ=η+Zᵢ with unknown lower endpoint η and Zᵢ exponential with known positive rate λ. Derive the distribution of n(m−η), where m is the sample minimum, and diagnose the ordinary bootstrap approximation to this endpoint error.
