02 MATH40005 SEVEN YEAR ABILITY UNITS
English review edition prepared on 4 October 2026 from a preserved source copy. It is a later presentation, not the historical study interface. Source checks and difficulty judgements describe the original material author's own process; they do not indicate Imperial College London endorsement.
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252 historical examination competency units
English translation of the source table, with wrapped cells reassembled. Each row retains its source unit, year, question, marks, topic codes and estimated time. These time estimates and workload ratings are training indicators, not times published by the university. Some source task descriptions are truncated; unresolved endings are explicitly labelled. Missing difficulty letters in the source are not invented. Standard squared symbols lost by PDF extraction are restored from the accompanying coverage workbook.
| Unit | Year | Question | Marks | Source difficulty | Specific competency task | Topic codes | Recognition cue | First key step | Method sequence | Computation / abstraction / integration | Estimated minutes | Source page |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| U001 | 2020 | Q1a | 3 | A×3 | Define a σ-algebra | K02 | The question involves σ-algebras, preimages, measurability or piecewise random variables. | Start from the three closure axioms or singleton preimages, rather than the range alone. | List axioms/preimage identities → check membership in F → establish measurability or non-measurability. | Low/Medium/Low | 4 | 1 |
| U002 | 2020 | Q1b | 3 | A×3 | Define a probability measure | K03 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. | Low/Low/Low | 4 | 1 |
| U003 | 2020 | Q1ci | 3 | B×3 | Use total probability to calculate the positive-test rate | K03,K08 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. Also combine with Bayes' theorem, diagnostic tests and posterior probabilities. | Low/Low/Medium | 4 | 1 |
| U004 | 2020 | Q1cii | 3 | B×3 | Use Bayes' theorem to find the probability of disease after a positive test | K08 | Prior rates and sensitivity/false-positive rates are given, and a posterior probability is requested. | Use total probability to form the denominator for the positive-test or observed event. | Define events → total-probability denominator → Bayes inversion → compare with the prior. | Low/Low/Low | 4 | 1 |
| U005 | 2020 | Q1d | 4 | B×4 | Enumerate ordered sums of three positive integers equal to 7 | K05,K06 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. Also combine with stars and bars, positive-integer partitions and multiset counting. | Medium/Low/Medium | 6 | 1 |
| U006 | 2020 | Q1e | 4 | C×1 + D×3 | Count ordered sums of n positive integers equal to a general k | K06 | The task concerns ordered positive-integer sums or multiset counting. | Determine whether 0 is allowed; for positive integers, place n-1 separators in the k-1 internal gaps. | Check k>=n → stars and bars → binomial coefficient → check boundary cases. | High/Low/High | 6 | 1 |
| U007 | 2020 | Q2a | 3 | C×3 | Prove that the given mapping is a discrete random variable | K02,K15 | The question involves σ-algebras, preimages, measurability or piecewise random variables. | Start from the three closure axioms or singleton preimages, rather than the range alone. | List axioms/preimage identities → check membership in F → establish measurability or non-measurability. Also combine with discrete transformations, mixtures and piecewise random variables. | Low/Medium/Medium | 4 | 1 |
| U008 | 2020 | Q2b | 4 | A×4 | Write a piecewise CDF from a discrete pmf | K11 | A discrete pmf or geometric-type mass is given, or a CDF on the whole real line is required. | State the support and accumulate mass over its points; use floor for noninteger inputs. | Check nonnegativity and total mass 1 → accumulate piecewise → check right-continuity and limits. | Medium/Low/Low | 6 | 1 |
| U009 | 2020 | Q2ci | 3 | A×2 + B×1 | Derive the CDF of a geometric distribution | K11,K14 | A discrete pmf or geometric-type mass is given, or a CDF on the whole real line is required. | State the support and accumulate mass over its points; use floor for noninteger inputs. | Check nonnegativity and total mass 1 → accumulate piecewise → check right-continuity and limits. Also combine with geometric waiting times, pattern waiting and discrete extremes. | Low/Low/Medium | 4 | 1 |
| U010 | 2020 | Q2cii | 4 | C×4 | Prove that the minimum of two independent geometric variables is geometric | K04,K14 | The question compares pairwise/joint independence, complementary events, or distributional conclusions under dependence. | Write the required product equalities, or construct a parity/perfect-dependence counterexample. | Pairwise intersection probabilities → triple intersection/factorisation → determine the level of independence. Also combine with geometric waiting times, pattern waiting and discrete extremes. | Medium/Low/Medium | 6 | 1 |
| U011 | 2020 | Q2d | 6 | A×1 + B×1 + C×1 + D×3 | Calculate the expected waiting time until the first HT pattern | K14,K23 | The task concerns first success, pattern waiting or extremes of geometric variables. | Use survival probabilities or state decomposition rather than enumerate all sequences. | Write P(X>k) or a state recurrence → use independence → identify geometric parameters or find expectations. Also combine with conditional expectation, total expectation and random mixtures. | High/High/High | 9 | 1 |
| U012 | 2020 | Q3a | 2 | A×2 | State the conditions for a valid probability density | K17 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. | Low/Low/Low | 3 | 1 |
| U013 | 2020 | Q3bi | 2 | A×2 | Assess and rescale 3x to obtain a valid density | K17 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. | Low/Low/Low | 3 | 1 |
| U014 | 2020 | Q3bii | 2 | A×2 | Assess a negative constant function and find a scaling constant | K17 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. | Low/Low/Low | 3 | 1 |
| U015 | 2020 | Q3biii | 2 | A×2 | Show that a piecewise function taking both signs cannot be scaled into a density | K17 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. | Low/Low/Low | 3 | 1 |
| U016 | 2020 | Q3ci | 3 | D×3 | Normalise a three-dimensional joint density to find its constant | K17,K22 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. Also combine with joint-density region integrals and geometric probability. | High/Low/High | 4 | 1 |
| U017 | 2020 | Q3cii | 3 | D×3 | Use three-dimensional LOTUS to calculate E[XYZ] | K20,K22,K25 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. Also combine with joint-density region integrals, geometric probability, joint moments, covariance and related quantities. | High/Low/High | 4 | 1 |
| U018 | 2020 | Q3di | 3 | A×3 | Define a partition of a sample space and give an example | K03 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. | Low/Low/Low | 4 | 1 |
| U019 | 2020 | Q3dii | 3 | B×3 | Prove the discrete law of total expectation | K03,K23 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. Also combine with conditional expectation, total expectation and random mixtures. | Low/Medium/Medium | 4 | 1 |
| U020 | 2020 | Q4ai | 3 | A×3 | Find the distribution of a normal sample mean | K24,K30 | A named distribution, its moments or standard properties must be identified. | Check the support, kernel and parameterisation convention. | Write the standard pmf/pdf/MGF → match parameters → check the mean, variance or domain. Also combine with sample means, sample variances, normal samples and χ² properties. | Low/Low/Medium | 4 | 2 |
| U021 | 2020 | Q4aii | 1 | A×1 | Use the unbiasedness of S² to find E[Z] | K30,K31 | The task involves sample means, sample variances, normal samples or chi-square properties. | Check identical-distribution/equal-mean assumptions, then use a sum-of-squares identity or normal pivot. | Decompose sums of squares → take expectations/use independence → normal-sample t/chi-square properties. Also combine with bias, unbiasedness, variance and MSE decomposition. | Low/Low/Medium | 2 | 2 |
| U022 | 2020 | Q4aiii | 1 | A×1 | Use Var(S²) to find Var(Z) | K30,K31 | The task involves sample means, sample variances, normal samples or chi-square properties. | Check identical-distribution/equal-mean assumptions, then use a sum-of-squares identity or normal pivot. | Decompose sums of squares → take expectations/use independence → normal-sample t/chi-square properties. Also combine with bias, unbiasedness, variance and MSE decomposition. | Low/Low/Medium | 2 | 2 |
| U023 | 2020 | Q4aiv | 1 | A×1 | Find the bias of Z as an estimator of σ² | K31 | An estimator's bias, variance or MSE is requested. | Calculate E and Var separately, then apply MSE=Var+Bias^2. | Expectation → bias → variance → MSE → check nonnegativity and order. | Low/Low/Low | 2 | 2 |
| U024 | 2020 | Q4av | 2 | A×2 | Find the mean squared error of Z | K31 | An estimator's bias, variance or MSE is requested. | Calculate E and Var separately, then apply MSE=Var+Bias^2. | Expectation → bias → variance → MSE → check nonnegativity and order. | Low/Low/Low | 3 | 2 |
| U025 | 2020 | Q4avi | 2 | B×2 | Scale Z to obtain a χ² distribution and state its degrees of freedom | K30 | The task involves sample means, sample variances, normal samples or chi-square properties. | Check identical-distribution/equal-mean assumptions, then use a sum-of-squares identity or normal pivot. | Decompose sums of squares → take expectations/use independence → normal-sample t/chi-square properties. | Low/Low/Low | 3 | 2 |
| U026 | 2020 | Q4avii | 2 | B×2 | Use independence properties of a normal sample to find Cov(Xbar,Z) | K27,K30 | The task concerns correlation, linear-combination correlation or covariance algebra. | Recover covariance from correlation, then calculate the variance of the new variable. | Covariance bilinearity → variance including cross terms → divide by standard deviations → check absolute value <=1. Also combine with sample means, sample variances, normal samples and χ² properties. | Low/Low/Medium | 3 | 2 |
| U027 | 2020 | Q4b | 4 | A×2 + B×2 | Prove Markov's inequality | K28 | Markov/Chebyshev bounds, distribution-free guarantees or range-based variance bounds are required. | Construct an indicator inequality or substitute the sample-mean variance into Chebyshev's inequality. | Pointwise inequality → expectation → solve for probability/sample size; check the direction of range and correlation bounds. | Medium/Low/Low | 6 | 2 |
| U028 | 2020 | Q4c | 4 | C×4 | Apply a Bonferroni correction to 100 genetic association tests | K41,K43 | A p-value must be defined or interpreted, or its null distribution proved. | Define the p-value using a prespecified notion of an extreme tail under H0. | Define the tail event → probability-integral transform using a continuous CDF → U(0,1) → P(p<=alpha)=alpha. Also combine with multiple testing and Bonferroni correction. | Medium/Low/Medium | 6 | 2 |
| U029 | 2020 | Q5a | 4 | A×4 | Prove that the sample mean minimises the sum of squared deviations | K29 | The best centre under squared or absolute loss is requested. | For squared loss, add and subtract E[X] and complete the square; for absolute loss, identify a median. | Decompose loss → nonnegative remainder → identify the mean/variance or median. | Medium/Low/Low | 6 | 2 |
| U030 | 2020 | Q5b | 7 | A×5 + B×2 | Derive the least-squares estimates for simple linear regression | K48 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. | High/Low/High | 10 | 2 |
| U031 | 2020 | Q5ci | 1 | A×1 | State the null hypothesis for a two-sample t-test | K40 | The task tests means or linear combinations from two independent samples. | State independence, normality and known/equal variance assumptions, then write the variances of both sample means. | H0/H1 → pooled-variance or known-variance standard error → statistic/degrees of freedom → p-value. | Low/Low/Low | 2 | 2 |
| U032 | 2020 | Q5cii | 2 | B×2 | State the normality and equal-variance assumptions for a two-sample t-test | K40 | The task tests means or linear combinations from two independent samples. | State independence, normality and known/equal variance assumptions, then write the variances of both sample means. | H0/H1 → pooled-variance or known-variance standard error → statistic/degrees of freedom → p-value. | Low/Low/Low | 3 | 2 |
| U033 | 2020 | Q5d | 6 | D×6 | Find the boundary MLE for U[0,θ] | K54 | The parameter determines the support of a Uniform or similar distribution. | Express the requirement that all observations lie in the support as parameter inequalities. | Zero/nonzero likelihood regions → feasible parameters → monotone or flat optimisation → unique/nonunique MLE. | High/Low/High | 9 | 2 |
| U034 | 2020 | Q6ai | 2 | A×2 | Construct a 90% confidence interval for a normal mean with known variance | K35 | A confidence interval for a normal mean with known population variance is requested. | Use the Z pivot and sigma/sqrt(n). | Write the pivot → obtain z quantiles → invert for mu → check symmetry. | Low/Low/Low | 3 | 2 |
| U035 | 2020 | Q6aii | 2 | A×1 + B×1 | Construct a 95% t interval for a normal mean with unknown variance | K36 | The population is normal with unknown variance and a relatively small sample. | Take the square root of s² to obtain s, and use a t pivot with n-1 degrees of freedom. | Confirm normality and unknown variance → t quantile → standard error → interval. | Low/Low/Low | 3 | 2 |
| U036 | 2020 | Q6aiii | 2 | C×2 | Use Chebyshev's inequality to construct a distribution-free 99% interval for a mean | K28,K45 | Markov/Chebyshev bounds, distribution-free guarantees or range-based variance bounds are required. | Construct an indicator inequality or substitute the sample-mean variance into Chebyshev's inequality. | Pointwise inequality → expectation → solve for probability/sample size; check the direction of range and correlation bounds. Also combine with sample-size determination using Chebyshev or normal approximations. | Low/Low/Medium | 3 | 2 |
| U037 | 2020 | Q6bi | 1 | B×1 | State the joint distribution assumptions for linear-regression errors | K48 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. | Low/Low/Low | 2 | 2 |
| U038 | 2020 | Q6bii | 4 | B×3 + C×1 | Assess fit using two residual plots and choose a transformation | K50 | Residual plots or R² are provided for diagnosis. | Inspect the central residual trend, then check constant variance and systematic patterns. | Identify curvature/funnelling/outliers → relate to model assumptions → propose transformations/extra terms/weights and recheck. | Medium/Low/Low | 6 | 2 |
| U039 | 2020 | Q6c | 3 | A×1 + C×2 | Find the posterior for an exponential likelihood with a Gamma prior | K55 | A likelihood and Gamma/Beta prior are given, and the posterior is required. | Retain the parameter-dependent kernel and use a consistent rate/scale convention. | Joint likelihood → multiply by prior → combine powers and exponentials → identify conjugate family and updated parameters. | Low/Medium/Low | 4 | 2 |
| U040 | 2020 | Q6d | 6 | D×6 | Bound sd(X) below using a correlation bound and the boundedness of Y | K27,K28,K33 | The task concerns correlation, linear-combination correlation or covariance algebra. | Recover covariance from correlation, then calculate the variance of the new variable. | Covariance bilinearity → variance including cross terms → divide by standard deviations → check absolute value <=1. Also combine with Markov/Chebyshev and bounded-variable variance bounds; nontrivial bounds using correlation and bounded variance. | High/Low/High | 9 | 2 |
| U041 | 2021 | Q1a | 2 | A×2 | Draw events in the unit square and assign probabilities by area | K01,K22 | A finite, equiprobable or multistage experiment requires Ω, F and P. | Check ordering and replacement assumptions, then write the Cartesian-product sample space. | Sample space → power-set event space → point/event probabilities → check total probability 1. Also combine with joint-density region integrals and geometric probability. | Low/Low/Medium | 3 | 2 |
| U042 | 2021 | Q1bi | 2 | A×2 | Write the sample space for seven die rolls | K01 | A finite, equiprobable or multistage experiment requires Ω, F and P. | Check ordering and replacement assumptions, then write the Cartesian-product sample space. | Sample space → power-set event space → point/event probabilities → check total probability 1. | Low/Low/Low | 3 | 2 |
| U043 | 2021 | Q1bii | 2 | A×2 | Find the cardinality of the power set of the sample space | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | Low/Low/Low | 3 | 3 |
| U044 | 2021 | Q1biii | 2 | B×2 | Construct the probability space for seven die rolls | K01,K03 | A finite, equiprobable or multistage experiment requires Ω, F and P. | Check ordering and replacement assumptions, then write the Cartesian-product sample space. | Sample space → power-set event space → point/event probabilities → check total probability 1. Also combine with probability measures, conditional probability, total probability and partitions. | Low/Low/Medium | 3 | 3 |
| U045 | 2021 | Q1c | 2 | A×2 | Prove that complements of independent events remain independent | K04 | The question compares pairwise/joint independence, complementary events, or distributional conclusions under dependence. | Write the required product equalities, or construct a parity/perfect-dependence counterexample. | Pairwise intersection probabilities → triple intersection/factorisation → determine the level of independence. | Low/Low/Low | 3 | 3 |
| U046 | 2021 | Q1di | 3 | B×3 | Count seven-letter words with distinct consonants and distinct vowels | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | Low/Low/Low | 4 | 3 |
| U047 | 2021 | Q1dii | 3 | B×1 + C×2 | Count seven-letter words allowing repeated vowels | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | Low/Low/Low | 4 | 3 |
| U048 | 2021 | Q1e | 2 | B×2 | Count multiset permutations of racing positions with repeated colours | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | Low/Low/Low | 3 | 3 |
| U049 | 2021 | Q1f | 2 | B×2 | Use the multiplication principle to count uniform combinations | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | Low/Low/Low | 3 | 3 |
| U050 | 2021 | Q2a | 3 | A×3 | Condition on the first die roll to analyse the maximum of two rolls [source task ending is truncated] | K03,K11,K14 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. Also combine with discrete-distribution recognition, pmf/CDF and support, geometric waiting times, pattern waiting and discrete extremes. | Low/Low/High | 4 | 3 |
| U051 | 2021 | Q2b | 6 | C×3 + D×3 | Use LOTUS and series to find the expectation of a function of a geometric variable | K14,K20 | The task concerns first success, pattern waiting or extremes of geometric variables. | Use survival probabilities or state decomposition rather than enumerate all sequences. | Write P(X>k) or a state recurrence → use independence → identify geometric parameters or find expectations. Also combine with continuous transformations using CDFs, LOTUS and density transformations. | High/Low/High | 9 | 3 |
| U052 | 2021 | Q2c | 4 | D×4 | Prove that integrating conditional distributions recovers the marginal distribution | K18,K23 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. Also combine with conditional expectation, total expectation and random mixtures. | High/High/High | 6 | 3 |
| U053 | 2021 | Q2d | 7 | C×4 + D×3 | Find the MGF of a Normal-Gamma scale mixture | K12,K23,K24 | The task involves PGFs/MGFs, identifying a sum's distribution or moment determinacy. | Derive the generating function from its definition; multiply for independent sums. | Defining integral/sum → algebra and domain → compare with standard generating functions → uniqueness. Also combine with conditional expectation, total expectation, random mixtures and Normal/Uniform/Exponential/Gamma/Beta distributions. | High/High/High | 10 | 3 |
| U054 | 2021 | Q3ai | 2 | A×2 | Find the CDF of a shifted Uniform variable | K20,K24 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. Also combine with Normal/Uniform/Exponential/Gamma/Beta distributions. | Low/Low/Medium | 3 | 3 |
| U055 | 2021 | Q3aii | 3 | A×3 | Prove that the shifted variable is still continuous | K17,K20 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. Also combine with continuous transformations using CDFs, LOTUS and density transformations. | Low/Low/Medium | 4 | 3 |
| U056 | 2021 | Q3aiii | 4 | A×2 + B×2 | Compare the mean and median of a shifted Uniform distribution | K24,K29 | A named distribution, its moments or standard properties must be identified. | Check the support, kernel and parameterisation convention. | Write the standard pmf/pdf/MGF → match parameters → check the mean, variance or domain. Also combine with means/medians as squared-loss/absolute-loss minimisers. | Medium/Low/Medium | 6 | 3 |
| U057 | 2021 | Q3b | 3 | B×3 | Explain when and why LOTUS is used | K20 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. | Low/Low/Low | 4 | 3 |
| U058 | 2021 | Q3ci | 1 | A×1 | Normalise a joint density to find its constant | K17 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. | Low/Low/Low | 2 | 3 |
| U059 | 2021 | Q3cii | 3 | A×3 | Find both marginal densities from a joint density | K18 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. | Low/Low/Low | 4 | 3 |
| U060 | 2021 | Q3ciii | 4 | A×2 + D×2 | Integrate over the region for P(X/Y≤0.5) | K22 | The joint support is triangular, curved or defined by multiple ordering constraints. | Draw or rewrite the support and state both integration orders. | Region boundaries → iterated integrals → check by reversing order → probabilities/moments. | High/Low/High | 6 | 3 |
| U061 | 2021 | Q4ai | 1 | A×1 | Prove that the second moment equals the squared mean plus the variance | K26 | The task requires a mean, variance or covariance identity for linear combinations. | Expand the linear combination using linearity of expectation and bilinearity of covariance. | Linearity of expectation → variance/covariance cross terms → substitute independence or known covariance. | Low/Low/Low | 2 | 3 |
| U062 | 2021 | Q4aii | 2 | A×2 | Prove that the sample second moment is an unbiased estimator | K31,K32 | An estimator's bias, variance or MSE is requested. | Calculate E and Var separately, then apply MSE=Var+Bias^2. | Expectation → bias → variance → MSE → check nonnegativity and order. Also combine with estimator construction and expectation/distribution/consistency properties. | Low/Low/Medium | 3 | 3 |
| U063 | 2021 | Q4aiii | 3 | A×1 + D×2 | Construct an unbiased estimator of (E[X])² | K31,K32 | An estimator's bias, variance or MSE is requested. | Calculate E and Var separately, then apply MSE=Var+Bias^2. | Expectation → bias → variance → MSE → check nonnegativity and order. Also combine with estimator construction and expectation/distribution/consistency properties. | High/Low/High | 4 | 3 |
| U064 | 2021 | Q4aiv | 2 | A×1 + C×1 | Calculate the realised estimate and note its possible negativity | K31,K32 | An estimator's bias, variance or MSE is requested. | Calculate E and Var separately, then apply MSE=Var+Bias^2. | Expectation → bias → variance → MSE → check nonnegativity and order. Also combine with estimator construction and expectation/distribution/consistency properties. | Low/Low/Medium | 3 | 4 |
| U065 | 2021 | Q4b | 6 | A×2 + B×1 + D×3 | Use Chebyshev's inequality to determine the sample size for average potato weight | K28,K45 | Markov/Chebyshev bounds, distribution-free guarantees or range-based variance bounds are required. | Construct an indicator inequality or substitute the sample-mean variance into Chebyshev's inequality. | Pointwise inequality → expectation → solve for probability/sample size; check the direction of range and correlation bounds. Also combine with sample-size determination using Chebyshev or normal approximations. | High/Low/High | 9 | 4 |
| U066 | 2021 | Q4ci | 1 | A×1 | Calculate the sample median | K46 | Raw data are given and a median, quartiles or outliers are requested. | Sort first and state the quartile convention. | Sort → median and lower/upper halves → IQR → Tukey fences → outliers. | Low/Low/Low | 2 | 4 |
| U067 | 2021 | Q4cii | 2 | A×2 | Calculate the lower and upper quartiles | K46 | Raw data are given and a median, quartiles or outliers are requested. | Sort first and state the quartile convention. | Sort → median and lower/upper halves → IQR → Tukey fences → outliers. | Low/Low/Low | 3 | 4 |
| U068 | 2021 | Q4ciii | 2 | A×2 | Identify outliers using Tukey's rule | K46 | Raw data are given and a median, quartiles or outliers are requested. | Sort first and state the quartile convention. | Sort → median and lower/upper halves → IQR → Tukey fences → outliers. | Low/Low/Low | 3 | 4 |
| U069 | 2021 | Q4civ | 1 | A×1 | Choose a boxplot to show quantiles and outliers | K47 | A scatterplot, boxplot, Q-Q plot or time-series plot must be chosen or interpreted. | Identify the variable type and objective: relationship, distribution, normality or time order. | Choose the graph → identify features to inspect → state a suitably qualified conclusion. | Low/Low/Low | 2 | 4 |
| U070 | 2021 | Q5a | 6 | A×2 + B×2 | Find the regular MLE for the θx^{θ-1} model | K53 | A differentiable likelihood is given and its MLE requested. | Write the full likelihood/log-likelihood, retaining parameter support and boundaries. | log L → score=0 → compare second derivatives and boundaries → estimator. | High/Low/High | 9 | 4 |
| U071 | 2021 | Q5bi | 1 | A×1 | State two-sided hypotheses about an unknown mean | K39 | The task is a one-sample mean hypothesis test. | State H0/H1 and choose z or t according to whether variance is known. | Standardise the statistic → degrees of freedom/tail → p-value/critical region → cautious conclusion. | Low/Low/Low | 2 | 4 |
| U072 | 2021 | Q5bii | 3 | A×1 + B×2 | Construct a one-sample t statistic and critical value | K39 | The task is a one-sample mean hypothesis test. | State H0/H1 and choose z or t according to whether variance is known. | Standardise the statistic → degrees of freedom/tail → p-value/critical region → cautious conclusion. | Low/Low/Low | 4 | 4 |
| U073 | 2021 | Q5biii | 2 | B×1 + C×1 | Draw a test conclusion from the statistic | K39 | The task is a one-sample mean hypothesis test. | State H0/H1 and choose z or t according to whether variance is known. | Standardise the statistic → degrees of freedom/tail → p-value/critical region → cautious conclusion. | Low/Low/Low | 3 | 4 |
| U074 | 2021 | Q5biv | 1 | A×1 | Construct the corresponding 99% t confidence interval | K36 | The population is normal with unknown variance and a relatively small sample. | Take the square root of s² to obtain s, and use a t pivot with n-1 degrees of freedom. | Confirm normality and unknown variance → t quantile → standard error → interval. | Low/Low/Low | 2 | 4 |
| U075 | 2021 | Q5c | 3 | B×1 + C×2 | Give an example of a two-sample t-test and state its assumptions | K40 | The task tests means or linear combinations from two independent samples. | State independence, normality and known/equal variance assumptions, then write the variances of both sample means. | H0/H1 → pooled-variance or known-variance standard error → statistic/degrees of freedom → p-value. | Low/Low/Low | 4 | 4 |
| U076 | 2021 | Q5d | 4 | B×1 + C×1 | Find the expected number of 200 p-values below 0.05 when all null hypotheses are true | K09,K41 | Independent yes/no events are summed into a count. | Define indicators and their success probabilities. | Individual Bernoulli variables → independent sum → Binomial parameters → check the mean. Also combine with p-value definition and interpretation, including U(0,1) under H0. | Medium/Low/Medium | 6 | 4 |
| U077 | 2021 | Q6ai | 1 | A×1 | Prove that the regression response variance is σ² | K48,K49 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. Also combine with variance, covariance and algebraic properties of regression estimators. | Low/Medium/Medium | 2 | 4 |
| U078 | 2021 | Q6aii | 2 | A×2 | Rewrite Sxy and express the slope estimator as a linear combination | K48,K49 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. Also combine with variance, covariance and algebraic properties of regression estimators. | Low/Medium/Medium | 3 | 4 |
| U079 | 2021 | Q6aiii | 4 | B×3 + C×1 | Calculate Cov(Yi,βhat1) | K27,K49 | The task concerns correlation, linear-combination correlation or covariance algebra. | Recover covariance from correlation, then calculate the variance of the new variable. | Covariance bilinearity → variance including cross terms → divide by standard deviations → check absolute value <=1. Also combine with variance, covariance and algebraic properties of regression estimators. | Medium/Medium/Medium | 6 | 4 |
| U080 | 2021 | Q6aiv | 3 | B×1 + D×2 | Give a linear-regression example and explain the predictive use of its coefficients | K48 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. | High/Low/High | 4 | 4 |
| U081 | 2021 | Q6b | 5 | A×2 + B×2 + C×1 | Find the posterior for a Geometric likelihood with a Beta prior | K55 | A likelihood and Gamma/Beta prior are given, and the posterior is required. | Retain the parameter-dependent kernel and use a consistent rate/scale convention. | Joint likelihood → multiply by prior → combine powers and exponentials → identify conjugate family and updated parameters. | Medium/Medium/Low | 8 | 4 |
| U082 | 2021 | Q6c | 2 | B×1 + C×1 | Identify linear-model misspecification from U-shaped residuals | K50 | Residual plots or R² are provided for diagnosis. | Inspect the central residual trend, then check constant variance and systematic patterns. | Identify curvature/funnelling/outliers → relate to model assumptions → propose transformations/extra terms/weights and recheck. | Low/Low/Low | 3 | 4 |
| U083 | 2021 | Q6di | 1 | A×1 | Identify Simpson's paradox | K51 | The prompt involves high correlation, aggregation reversal or a causal claim. | Separate sample association from causation and examine trends, confounding and group weights. | Describe correlation → explain why it does not establish causation → confounding/time trends/Simpson mechanism → additional evidence needed. | Low/Low/Low | 2 | 4 |
| U084 | 2021 | Q6dii | 1 | D×1 | Explain the conditions producing Simpson's paradox | K51 | The prompt involves high correlation, aggregation reversal or a causal claim. | Separate sample association from causation and examine trends, confounding and group weights. | Describe correlation → explain why it does not establish causation → confounding/time trends/Simpson mechanism → additional evidence needed. | High/Low/High | 2 | 4 |
| U085 | 2021 | Q6diii | 1 | D×1 | Explain the reversal by stratifying patients by risk | K51 | The prompt involves high correlation, aggregation reversal or a causal claim. | Separate sample association from causation and examine trends, confounding and group weights. | Describe correlation → explain why it does not establish causation → confounding/time trends/Simpson mechanism → additional evidence needed. | High/Low/High | 2 | 4 |
| U086 | 2022 | Q1ai | 2 | B×2 | Give a nontrivial σ-algebra on a four-point sample space | K02 | The question involves σ-algebras, preimages, measurability or piecewise random variables. | Start from the three closure axioms or singleton preimages, rather than the range alone. | List axioms/preimage identities → check membership in F → establish measurability or non-measurability. | Low/Medium/Low | 3 | 4 |
| U087 | 2022 | Q1aii | 3 | B×3 | Construct a discrete random variable and prove measurability | K02,K15 | The question involves σ-algebras, preimages, measurability or piecewise random variables. | Start from the three closure axioms or singleton preimages, rather than the range alone. | List axioms/preimage identities → check membership in F → establish measurability or non-measurability. Also combine with discrete transformations, mixtures and piecewise random variables. | Low/Medium/Medium | 4 | 4 |
| U088 | 2022 | Q1aiii | 5 | B×2 + C×3 | Construct a discrete uniform variable and a probability measure | K01,K02,K03,K11 | A finite, equiprobable or multistage experiment requires Ω, F and P. | Check ordering and replacement assumptions, then write the Cartesian-product sample space. | Sample space → power-set event space → point/event probabilities → check total probability 1. Also combine with σ-algebras, preimages, measurability, random-variable definitions, probability measures, conditional/total probability, partitions and discrete pmf/CDF/support. | Medium/Medium/High | 8 | 5 |
| U089 | 2022 | Q1bi | 3 | B×3 | Prove a combinatorial identity using the binomial theorem or induction | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | Low/Low/Low | 4 | 5 |
| U090 | 2022 | Q1bii | 7 | D×7 | Give a story proof of a combinatorial identity | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | High/Low/High | 10 | 5 |
| U091 | 2022 | Q2a | 3 | A×3 | Prove the bilinear covariance formula for linear combinations | K27 | The task concerns correlation, linear-combination correlation or covariance algebra. | Recover covariance from correlation, then calculate the variance of the new variable. | Covariance bilinearity → variance including cross terms → divide by standard deviations → check absolute value <=1. | Low/Low/Low | 4 | 5 |
| U092 | 2022 | Q2bi | 5 | A×5 | Find the expectation and variance of a linear combination | K26,K27 | The task requires a mean, variance or covariance identity for linear combinations. | Expand the linear combination using linearity of expectation and bilinearity of covariance. | Linearity of expectation → variance/covariance cross terms → substitute independence or known covariance. Also combine with covariance, correlation and linear-combination algebra. | Medium/Low/Medium | 8 | 5 |
| U093 | 2022 | Q2bii | 4 | A×4 | Calculate the covariance of two linear combinations | K27 | The task concerns correlation, linear-combination correlation or covariance algebra. | Recover covariance from correlation, then calculate the variance of the new variable. | Covariance bilinearity → variance including cross terms → divide by standard deviations → check absolute value <=1. | Medium/Low/Low | 6 | 5 |
| U094 | 2022 | Q2ci | 7 | A×1 + C×6 | Find and verify the density of Z=XY when X is continuous and Y is discrete | K15,K20,K23 | Random variables are defined piecewise over events, or a continuous variable is multiplied by a discrete one. | For piecewise variables check singleton preimages; for mixtures first condition on the discrete variable. | Range/preimages or conditional CDF → sum the mixture → differentiate/normalise. Also combine with continuous transformations using CDFs, LOTUS and densities; conditional/total expectation and mixtures. | High/High/High | 10 | 5 |
| U095 | 2022 | Q2cii | 1 | A×1 | Find the mixed CDF of Z=XY when zero is possible | K15,K20 | Random variables are defined piecewise over events, or a continuous variable is multiplied by a discrete one. | For piecewise variables check singleton preimages; for mixtures first condition on the discrete variable. | Range/preimages or conditional CDF → sum the mixture → differentiate/normalise. Also combine with continuous transformations using CDFs, LOTUS and density transformations. | Low/Low/Medium | 2 | 5 |
| U096 | 2022 | Q3a | 8 | A×6 + B×2 | Check reported Bayes conclusions at low/high disease prevalence and identify the issue [source task ending is truncated] | K08 | Prior rates and sensitivity/false-positive rates are given, and a posterior probability is requested. | Use total probability to form the denominator for the positive-test or observed event. | Define events → total-probability denominator → Bayes inversion → compare with the prior. | High/Low/High | 12 | 5 |
| U097 | 2022 | Q3b | 6 | A×1 + D×5 | Find probabilities of conditional events involving Uniform maxima/minima | K03,K21,K22 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. Also combine with order statistics (sample maxima/minima), joint-density region integration and geometric probability. | High/Low/High | 9 | 5 |
| U098 | 2022 | Q3ci | 1 | A×1 | Normalise a joint density to find its constant | K17 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. | Low/Low/Low | 2 | 5 |
| U099 | 2022 | Q3cii | 2 | A×2 | Find the marginal density of X | K18 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. | Low/Low/Low | 3 | 5 |
| U100 | 2022 | Q3ciii | 3 | B×3 | Find E(Y|X=x) and state its domain | K18,K23 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. Also combine with conditional expectation, total expectation and random mixtures. | Low/Medium/Medium | 4 | 5 |
| U101 | 2022 | Q4ai | 2 | A×2 | Find the bias of the given estimator | K31 | An estimator's bias, variance or MSE is requested. | Calculate E and Var separately, then apply MSE=Var+Bias^2. | Expectation → bias → variance → MSE → check nonnegativity and order. | Low/Low/Low | 3 | 5 |
| U102 | 2022 | Q4aii | 3 | A×3 | Find the estimator's MSE | K31 | An estimator's bias, variance or MSE is requested. | Calculate E and Var separately, then apply MSE=Var+Bias^2. | Expectation → bias → variance → MSE → check nonnegativity and order. | Low/Low/Low | 4 | 5 |
| U103 | 2022 | Q4b | 6 | A×2 + B×4 | Choose a sample size using the Bernoulli variance bound and Chebyshev's inequality | K09,K28,K45 | Independent yes/no events are summed into a count. | Define indicators and their success probabilities. | Individual Bernoulli variables → independent sum → Binomial parameters → check the mean. Also combine with Markov/Chebyshev and bounded-variable variance bounds; sample sizes from Chebyshev/normal approximations. | High/Low/High | 9 | 5 |
| U104 | 2022 | Q4c | 3 | C×2 + D×1 | Prove the median formula for Beta(α,1) | K20,K24,K29 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. Also combine with Normal/Uniform/Exponential/Gamma/Beta distributions and means/medians as squared-loss/absolute-loss minimisers. | High/Low/High | 4 | 5 |
| U105 | 2022 | Q4di | 4 | A×4 | Construct a 99% t interval for a normal mean with unknown variance | K36 | The population is normal with unknown variance and a relatively small sample. | Take the square root of s² to obtain s, and use a t pivot with n-1 degrees of freedom. | Confirm normality and unknown variance → t quantile → standard error → interval. | Medium/Low/Low | 6 | 5 |
| U106 | 2022 | Q4dii | 2 | D×2 | Use a confidence interval to assess whether a stock-price mean is positive | K38 | A realised confidence interval must be interpreted. | Distinguish a fixed parameter from the random interval before sampling. | Repeated-sampling coverage → fixed realised interval either includes the parameter or does not → reject a posterior-probability interpretation. | High/Low/High | 3 | 5 |
| U107 | 2022 | Q5ai | 2 | A×2 | Define Type I and Type II errors | K42 | Type I/II errors, significance level and power are at issue. | State the error event separately when H0 is true and when it is false. | Type I=P(reject|H0 true) → Type II=P(fail to reject|alternative) → changing alpha and the power trade-off. | Low/Low/Low | 3 | 5 |
| U108 | 2022 | Q5aii | 2 | B×1 + C×1 | Discuss the trade-off between power and the two error types | K42 | Type I/II errors, significance level and power are at issue. | State the error event separately when H0 is true and when it is false. | Type I=P(reject|H0 true) → Type II=P(fail to reject|alternative) → changing alpha and the power trade-off. | Low/Low/Low | 3 | 5 |
| U109 | 2022 | Q5aiii | 2 | D×2 | Interpret the two error types in a court setting | K42 | Type I/II errors, significance level and power are at issue. | State the error event separately when H0 is true and when it is false. | Type I=P(reject|H0 true) → Type II=P(fail to reject|alternative) → changing alpha and the power trade-off. | High/Low/High | 3 | 6 |
| U110 | 2022 | Q5bi | 2 | B×2 | State hypotheses for a one-sided mean test | K39 | The task is a one-sample mean hypothesis test. | State H0/H1 and choose z or t according to whether variance is known. | Standardise the statistic → degrees of freedom/tail → p-value/critical region → cautious conclusion. | Low/Low/Low | 3 | 6 |
| U111 | 2022 | Q5bii | 4 | [initial category absent in source] ×4 | Perform a one-sample z-test with known variance and draw a conclusion | K39 | The task is a one-sample mean hypothesis test. | State H0/H1 and choose z or t according to whether variance is known. | Standardise the statistic → degrees of freedom/tail → p-value/critical region → cautious conclusion. | Medium/Low/Low | 6 | 6 |
| U112 | 2022 | Q5ci | 2 | [initial category absent in source] ×2 | Assess whether the two smallest p-values are significant | K41,K43 | A p-value must be defined or interpreted, or its null distribution proved. | Define the p-value using a prespecified notion of an extreme tail under H0. | Define the tail event → probability-integral transform using a continuous CDF → U(0,1) → P(p<=alpha)=alpha. Also combine with multiple testing and Bonferroni correction. | Low/Low/Medium | 3 | 6 |
| U113 | 2022 | Q5cii | 2 | [initial category absent in source] ×2 | Adjust the threshold so that exactly one finding is significant | K43 | Multiple simultaneous tests require control of the overall error rate. | Count the tests and divide the family significance level by m. | Bonferroni threshold alpha/m → compare all p-values → explain family-wise error-rate control. | Low/Low/Low | 3 | 6 |
| U114 | 2022 | Q5d | 4 | [initial category absent in source] ×1 + B×1 | Calculate the sample correlation from summary quantities | K27 | The task concerns correlation, linear-combination correlation or covariance algebra. | Recover covariance from correlation, then calculate the variance of the new variable. | Covariance bilinearity → variance including cross terms → divide by standard deviations → check absolute value <=1. | Medium/Low/Low | 6 | 6 |
| U115 | 2022 | Q6a | 6 | [initial category absent in source] ×6 | Find the MLE of the location parameter of a lognormal distribution | K53 | A differentiable likelihood is given and its MLE requested. | Write the full likelihood/log-likelihood, retaining parameter support and boundaries. | log L → score=0 → compare second derivatives and boundaries → estimator. | High/Low/High | 9 | 6 |
| U116 | 2022 | Q6bi | 1 | [initial category absent in source] ×1 | Determine whether the given sample is a bootstrap sample | K52 | A bootstrap sample must be assessed or a bootstrap interval constructed. | Check replacement and equal sample size; for intervals take quantiles of the resampled statistic. | Resample B times → calculate the statistic each time → empirical distribution → 2.5%/97.5% endpoints. | Low/Low/Low | 2 | 6 |
| U117 | 2022 | Q6bii | 1 | [initial category absent in source] ×1 | Reject a proposed bootstrap sample containing values absent from the original sample | K52 | A bootstrap sample must be assessed or a bootstrap interval constructed. | Check replacement and equal sample size; for intervals take quantiles of the resampled statistic. | Resample B times → calculate the statistic each time → empirical distribution → 2.5%/97.5% endpoints. | Low/Low/Low | 2 | 6 |
| U118 | 2022 | Q6c | 2 | [initial category absent in source] ×2 | Use R² and residual plots to identify model misspecification | K50 | Residual plots or R² are provided for diagnosis. | Inspect the central residual trend, then check constant variance and systematic patterns. | Identify curvature/funnelling/outliers → relate to model assumptions → propose transformations/extra terms/weights and recheck. | Low/Low/Low | 3 | 6 |
| U119 | 2022 | Q6d | 4 | [initial category absent in source] ×4 | Check coefficients using the fact that the regression line passes through (xbar,ybar) | K48 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. | Medium/Low/Low | 6 | 6 |
| U120 | 2022 | Q6e | 6 | [initial category absent in source] ×2 + B×3 | Update a Gamma prior using a specified nonstandard likelihood | K55 | A likelihood and Gamma/Beta prior are given, and the posterior is required. | Retain the parameter-dependent kernel and use a consistent rate/scale convention. | Joint likelihood → multiply by prior → combine powers and exponentials → identify conjugate family and updated parameters. | High/High/High | 9 | 6 |
| U121 | 2023 | Q1a | 5 | [initial category absent in source] ×5 | Find the probability that at least two of 20 people share a birthday week | K07 | The task involves birthday collisions, at least one pair or exactly one pair. | Use the complement for at least one pair; for exactly one pair select the pair and exclude all other collisions. | Independent uniform assumptions → choose the collision structure → allocate the remaining people without replacement → divide by the total number of outcomes. | Medium/Low/Low | 8 | 6 |
| U122 | 2023 | Q1b | 5 | [initial category absent in source] ×1 + B×4 | Find the probability that exactly one pair among n people shares a birthday | K05,K07 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. Also combine with birthday/collision problems and complementary-event counting. | Medium/Low/Medium | 8 | 6 |
| U123 | 2023 | Q1c | 5 | [initial category absent in source] ×5 | Find a Bayes posterior for influenza/measles given a rash | K08 | Prior rates and sensitivity/false-positive rates are given, and a posterior probability is requested. | Use total probability to form the denominator for the positive-test or observed event. | Define events → total-probability denominator → Bayes inversion → compare with the prior. | Medium/Low/Low | 8 | 6 |
| U124 | 2023 | Q1d | 5 | [initial category absent in source] ×1 + C×4 | Prove measurability of discrete random variables joined piecewise over events | K02,K15 | The question involves σ-algebras, preimages, measurability or piecewise random variables. | Start from the three closure axioms or singleton preimages, rather than the range alone. | List axioms/preimage identities → check membership in F → establish measurability or non-measurability. Also combine with discrete transformations, mixtures and piecewise random variables. | Medium/Medium/Medium | 8 | 6 |
| U125 | 2023 | Q2ai | 2 | [initial category absent in source] ×2 | Prove an indicator-function identity involving preimages | K02 | The question involves σ-algebras, preimages, measurability or piecewise random variables. | Start from the three closure axioms or singleton preimages, rather than the range alone. | List axioms/preimage identities → check membership in F → establish measurability or non-measurability. | Low/Medium/Low | 3 | 6 |
| U126 | 2023 | Q2aii | 6 | [initial category absent in source] ×2 + C×4 | Prove that the preimage family of a σ-algebra is a σ-algebra | K02 | The question involves σ-algebras, preimages, measurability or piecewise random variables. | Start from the three closure axioms or singleton preimages, rather than the range alone. | List axioms/preimage identities → check membership in F → establish measurability or non-measurability. | High/High/High | 9 | 6 |
| U127 | 2023 | Q2b | 6 | [initial category absent in source] ×6 | Find the tail probability of a Uniform minimum with a Poisson random sample size | K13,K21,K23 | The task involves Poisson sums, conditional allocation, random sums or compound Poisson variables. | Check independence; for a random sum condition first on the count N. | Condition on N/use convolution or an MGF → Poisson moments/exponential series → identify the distribution or moments. Also combine with order statistics (sample maxima/minima), conditional/total expectation and random mixtures. | High/High/High | 9 | 6 |
| U128 | 2023 | Q2c | 6 | D×6 | Find a region probability for two independent standard normal variables | K22,K24 | The joint support is triangular, curved or defined by multiple ordering constraints. | Draw or rewrite the support and state both integration orders. | Region boundaries → iterated integrals → check by reversing order → probabilities/moments. Also combine with Normal/Uniform/Exponential/Gamma/Beta distributions. | High/Low/High | 9 | 6 |
| U129 | 2023 | Q3a | 3 | A×2 + C×1 | Explain why a sum of Poisson variables need not be Poisson without independence | K04,K13 | The question compares pairwise/joint independence, complementary events, or distributional conclusions under dependence. | Write the required product equalities, or construct a parity/perfect-dependence counterexample. | Pairwise intersection probabilities → triple intersection/factorisation → determine the level of independence. Also combine with Poisson superposition, random sums and compound Poisson distributions. | Low/Low/Medium | 4 | 6 |
| U130 | 2023 | Q3bi | 1 | A×1 | Normalise a joint density to find its constant | K17 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. | Low/Low/Low | 2 | 6 |
| U131 | 2023 | Q3bii | 2 | A×2 | Find both marginal densities | K18 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. | Low/Low/Low | 3 | 7 |
| U132 | 2023 | Q3biii | 1 | A×1 | Determine that X and Y are not independent | K19 | The question asks whether continuous variables are independent. | Check both density factorisation and whether the support is a Cartesian product. | Find marginals → compare fXY with fXfY, including zero-density regions → decisive counterexample/conclusion. | Low/Low/Low | 2 | 7 |
| U133 | 2023 | Q3biv | 4 | A×2 + B×2 | Find the conditional CDF of Y given X=x | K18 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. | Medium/Low/Low | 6 | 7 |
| U134 | 2023 | Q3ci | 3 | A×3 | Find the CDF and density of X^4 | K20,K24 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. Also combine with Normal/Uniform/Exponential/Gamma/Beta distributions. | Low/Low/Medium | 4 | 7 |
| U135 | 2023 | Q3cii | 2 | A×2 | Calculate E[X^4] | K20,K24 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. Also combine with Normal/Uniform/Exponential/Gamma/Beta distributions. | Low/Low/Medium | 3 | 7 |
| U136 | 2023 | Q3d | 4 | B×4 | Use MGFs to prove the distribution of a sum of independent Binomial variables | K09,K12 | Independent yes/no events are summed into a count. | Define indicators and their success probabilities. | Individual Bernoulli variables → independent sum → Binomial parameters → check the mean. Also combine with probability/moment generating functions and distributional uniqueness. | Medium/Medium/Medium | 6 | 7 |
| U137 | 2023 | Q4a | 3 | A×3 | Prove MSE=Bias²+Variance | K31 | An estimator's bias, variance or MSE is requested. | Calculate E and Var separately, then apply MSE=Var+Bias^2. | Expectation → bias → variance → MSE → check nonnegativity and order. | Low/Low/Low | 4 | 7 |
| U138 | 2023 | Q4b | 5 | A×1 + B×4 | Use Chebyshev's inequality to determine a polling sample size | K09,K28,K45 | Independent yes/no events are summed into a count. | Define indicators and their success probabilities. | Individual Bernoulli variables → independent sum → Binomial parameters → check the mean. Also combine with Markov/Chebyshev and bounded-variable variance bounds; sample sizes from Chebyshev/normal approximations. | Medium/Low/High | 8 | 7 |
| U139 | 2023 | Q4ci | 4 | A×3 + C×1 | Construct a 97% interval for a normal mean with known variance | K35 | A confidence interval for a normal mean with known population variance is requested. | Use the Z pivot and sigma/sqrt(n). | Write the pivot → obtain z quantiles → invert for mu → check symmetry. | Medium/Low/Low | 6 | 7 |
| U140 | 2023 | Q4cii | 3 | C×3 | Interpret a given 99% confidence interval | K38 | A realised confidence interval must be interpreted. | Distinguish a fixed parameter from the random interval before sampling. | Repeated-sampling coverage → fixed realised interval either includes the parameter or does not → reject a posterior-probability interpretation. | Low/Low/Low | 4 | 7 |
| U141 | 2023 | Q4di | 2 | A×2 | Calculate the lower and upper quartiles | K46 | Raw data are given and a median, quartiles or outliers are requested. | Sort first and state the quartile convention. | Sort → median and lower/upper halves → IQR → Tukey fences → outliers. | Low/Low/Low | 3 | 7 |
| U142 | 2023 | Q4dii | 3 | A×2 + B×1 | Identify outliers using Tukey's criterion | K46 | Raw data are given and a median, quartiles or outliers are requested. | Sort first and state the quartile convention. | Sort → median and lower/upper halves → IQR → Tukey fences → outliers. | Low/Low/Low | 4 | 7 |
| U143 | 2023 | Q5ai | 2 | B×2 | Define a p-value | K41 | A p-value must be defined or interpreted, or its null distribution proved. | Define the p-value using a prespecified notion of an extreme tail under H0. | Define the tail event → probability-integral transform using a continuous CDF → U(0,1) → P(p<=alpha)=alpha. | Low/Low/Low | 3 | 7 |
| U144 | 2023 | Q5aii | 3 | C×3 | Explain the meaning of p=0.06 for the experiment | K41 | A p-value must be defined or interpreted, or its null distribution proved. | Define the p-value using a prespecified notion of an extreme tail under H0. | Define the tail event → probability-integral transform using a continuous CDF → U(0,1) → P(p<=alpha)=alpha. | Low/Low/Low | 4 | 7 |
| U145 | 2023 | Q5b | 6 | A×2 + B×1 + D×3 | Test whether the sum of two normal population means equals 360 | K40 | The task tests means or linear combinations from two independent samples. | State independence, normality and known/equal variance assumptions, then write the variances of both sample means. | H0/H1 → pooled-variance or known-variance standard error → statistic/degrees of freedom → p-value. | High/Low/High | 9 | 7 |
| U146 | 2023 | Q5c | 6 | B×2 + C×1 + D×3 | Calculate the correlation between Y and X+3Y | K27 | The task concerns correlation, linear-combination correlation or covariance algebra. | Recover covariance from correlation, then calculate the variance of the new variable. | Covariance bilinearity → variance including cross terms → divide by standard deviations → check absolute value <=1. | High/Low/High | 9 | 7 |
| U147 | 2023 | Q5d | 3 | B×1 + D×2 | Identify and explain Simpson's paradox in salary data | K51 | The prompt involves high correlation, aggregation reversal or a causal claim. | Separate sample association from causation and examine trends, confounding and group weights. | Describe correlation → explain why it does not establish causation → confounding/time trends/Simpson mechanism → additional evidence needed. | High/Low/High | 4 | 7 |
| U148 | 2023 | Q6a | 6 | A×2 + B×1 + C×1 + D×2 | Find the MLE of the geometric distribution parameter | K53 | A differentiable likelihood is given and its MLE requested. | Write the full likelihood/log-likelihood, retaining parameter support and boundaries. | log L → score=0 → compare second derivatives and boundaries → estimator. | High/Low/High | 9 | 7 |
| U149 | 2023 | Q6b | 6 | A×4 + B×2 | Derive OLS coefficients for simple linear regression with an intercept | K48 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. | High/Low/High | 9 | 7 |
| U150 | 2023 | Q6c | 3 | B×1 + D×2 | Diagnose residual plots and suggest variable transformations | K50 | Residual plots or R² are provided for diagnosis. | Inspect the central residual trend, then check constant variance and systematic patterns. | Identify curvature/funnelling/outliers → relate to model assumptions → propose transformations/extra terms/weights and recheck. | High/Low/High | 4 | 7 |
| U151 | 2023 | Q6di | 1 | A×1 | Choose a histogram for a continuous univariate distribution | K47 | A scatterplot, boxplot, Q-Q plot or time-series plot must be chosen or interpreted. | Identify the variable type and objective: relationship, distribution, normality or time order. | Choose the graph → identify features to inspect → state a suitably qualified conclusion. | Low/Low/Low | 2 | 7 |
| U152 | 2023 | Q6dii | 1 | A×1 | Choose a Q-Q plot to assess normality | K47 | A scatterplot, boxplot, Q-Q plot or time-series plot must be chosen or interpreted. | Identify the variable type and objective: relationship, distribution, normality or time order. | Choose the graph → identify features to inspect → state a suitably qualified conclusion. | Low/Low/Low | 2 | 7 |
| U153 | 2023 | Q6diii | 1 | A×1 | Choose a bar or pie chart for categorical proportions | K47 | A scatterplot, boxplot, Q-Q plot or time-series plot must be chosen or interpreted. | Identify the variable type and objective: relationship, distribution, normality or time order. | Choose the graph → identify features to inspect → state a suitably qualified conclusion. | Low/Low/Low | 2 | 7 |
| U154 | 2023 | Q6div | 1 | A×1 | Choose a time-series plot for three years of stock prices | K47 | A scatterplot, boxplot, Q-Q plot or time-series plot must be chosen or interpreted. | Identify the variable type and objective: relationship, distribution, normality or time order. | Choose the graph → identify features to inspect → state a suitably qualified conclusion. | Low/Low/Low | 2 | 8 |
| U155 | 2023 | Q6dv | 1 | A×1 | Choose a boxplot for stock-price quantiles | K47 | A scatterplot, boxplot, Q-Q plot or time-series plot must be chosen or interpreted. | Identify the variable type and objective: relationship, distribution, normality or time order. | Choose the graph → identify features to inspect → state a suitably qualified conclusion. | Low/Low/Low | 2 | 8 |
| U156 | 2024 | Q1a | 4 | B×4 | Distribute 50 identical footballs among 20 people, with at least 1 each | K06 | The task concerns ordered positive-integer sums or multiset counting. | Determine whether 0 is allowed; for positive integers, place n-1 separators in the k-1 internal gaps. | Check k>=n → stars and bars → binomial coefficient → check boundary cases. | Medium/Low/Low | 6 | 8 |
| U157 | 2024 | Q1bi | 4 | A×4 | Find the probability that a random arrangement of repeated letters is BOO | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | Medium/Low/Low | 6 | 8 |
| U158 | 2024 | Q1bii | 4 | A×4 | Find the probability that three E's are adjacent in an arrangement with repeated letters | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | Medium/Low/Low | 6 | 8 |
| U159 | 2024 | Q1c | 4 | A×4 | Use Bayes' theorem after a positive breast-cancer screening result | K08 | Prior rates and sensitivity/false-positive rates are given, and a posterior probability is requested. | Use total probability to form the denominator for the positive-test or observed event. | Define events → total-probability denominator → Bayes inversion → compare with the prior. | Medium/Low/Low | 6 | 8 |
| U160 | 2024 | Q1d | 4 | A×4 | Express P(X>x,Y>y) using the joint CDF | K03,K18 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. Also combine with marginal/conditional densities and distribution functions of joint distributions. | Medium/Low/Medium | 6 | 8 |
| U161 | 2024 | Q2a | 8 | A×2 + C×6 | Find the PGF of the initial same-face run length for a biased coin | K12,K14,K23 | The task involves PGFs/MGFs, identifying a sum's distribution or moment determinacy. | Derive the generating function from its definition; multiply for independent sums. | Defining integral/sum → algebra and domain → compare with standard generating functions → uniqueness. Also combine with geometric waiting times, pattern waiting, discrete extremes, conditional/total expectation and random mixtures. | High/High/High | 12 | 8 |
| U162 | 2024 | Q2bi | 2 | A×2 | Verify a joint density on a triangular region | K17 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. | Low/Low/Low | 3 | 8 |
| U163 | 2024 | Q2bii | 2 | A×2 | Find the marginal density of X | K18 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. | Low/Low/Low | 3 | 8 |
| U164 | 2024 | Q2biii | 2 | A×2 | Find the marginal density of Y | K18 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. | Low/Low/Low | 3 | 8 |
| U165 | 2024 | Q2biv | 1 | B×1 | Find E[X] | K20 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. | Low/Low/Low | 2 | 8 |
| U166 | 2024 | Q2bv | 1 | B×1 | Find E[Y] | K20 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. | Low/Low/Low | 2 | 8 |
| U167 | 2024 | Q2bvi | 4 | B×4 | Find Cov(X,Y) | K25,K27 | A joint distribution is given and E[XY], covariance or joint moments are required. | Write Cov=E[XY]-E[X]E[Y], using the correct support throughout. | Marginal means → joint LOTUS → subtract → check sign and magnitude. Also combine with covariance, correlation and linear-combination algebra. | Medium/Low/Medium | 6 | 8 |
| U168 | 2024 | Q3a | 6 | B×2 + D×4 | Prove that independent Poisson counts, conditional on their sum, have a Binomial distribution | K10,K13,K16 | A fixed-size sample is drawn without replacement from a population with fixed success/failure counts. | Identify the hypergeometric distribution and specify N, K and n. | Combinatorial numerator/denominator → exact probability or tail probability → check support. Also combine with Poisson superposition, random sums, compound Poisson distributions, discrete conditional distributions, Poisson splitting and thinning. | High/Low/High | 9 | 8 |
| U169 | 2024 | Q3bi | 4 | B×3 + C×1 | Derive the first and second moments of a Poisson variable | K12,K26 | The task involves PGFs/MGFs, identifying a sum's distribution or moment determinacy. | Derive the generating function from its definition; multiply for independent sums. | Defining integral/sum → algebra and domain → compare with standard generating functions → uniqueness. Also combine with expectation/variance identities and linear combinations. | Medium/Medium/Medium | 6 | 8 |
| U170 | 2024 | Q3bii | 5 | C×2 + D×3 | Find the expectation of a compound Poisson sum | K13,K23,K26 | The task involves Poisson sums, conditional allocation, random sums or compound Poisson variables. | Check independence; for a random sum condition first on the count N. | Condition on N/use convolution or an MGF → Poisson moments/exponential series → identify the distribution or moments. Also combine with conditional/total expectation, random mixtures, expectation/variance identities and linear combinations. | High/High/High | 8 | 8 |
| U171 | 2024 | Q3biii | 5 | D×5 | Find the variance of a compound Poisson sum | K13,K23,K26 | The task involves Poisson sums, conditional allocation, random sums or compound Poisson variables. | Check independence; for a random sum condition first on the count N. | Condition on N/use convolution or an MGF → Poisson moments/exponential series → identify the distribution or moments. Also combine with conditional/total expectation, random mixtures, expectation/variance identities and linear combinations. | High/High/High | 8 | 8 |
| U172 | 2024 | Q4a | 4 | A×4 | Prove that variance is the minimum expected squared loss | K29 | The best centre under squared or absolute loss is requested. | For squared loss, add and subtract E[X] and complete the square; for absolute loss, identify a median. | Decompose loss → nonnegative remainder → identify the mean/variance or median. | Medium/Low/Low | 6 | 8 |
| U173 | 2024 | Q4b | 2 | A×2 | Calculate the sample median and IQR | K46 | Raw data are given and a median, quartiles or outliers are requested. | Sort first and state the quartile convention. | Sort → median and lower/upper halves → IQR → Tukey fences → outliers. | Low/Low/Low | 3 | 8 |
| U174 | 2024 | Q4ci | 4 | A×4 | Construct a 99% t interval for a normal mean with unknown variance | K36 | The population is normal with unknown variance and a relatively small sample. | Take the square root of s² to obtain s, and use a t pivot with n-1 degrees of freedom. | Confirm normality and unknown variance → t quantile → standard error → interval. | Medium/Low/Low | 6 | 8 |
| U175 | 2024 | Q4cii | 4 | A×3 + B×1 | Construct a 95% χ² interval for a normal variance | K37 | A confidence interval for a normal population variance is required. | Write (n-1)S²/sigma²~chi-square and note the reversal of endpoint order when inverting. | Chi-square pivot → two tail quantiles → solve for sigma² → verify lower/upper endpoints. | Medium/Low/Low | 6 | 8 |
| U176 | 2024 | Q4d | 6 | C×1 + D×5 | Find the CDF of a Uniform maximum and an unbiased estimator of θ | K21,K31,K32 | The task involves sample maxima, minima or order statistics. | For the maximum CDF, require every observation to be below the threshold; use the survival function for the minimum. | Intersection event → independence product → piecewise CDF → optionally differentiate for a density. Also combine with bias, unbiasedness, variance, MSE decomposition and estimator expectation/distribution/consistency properties. | High/Low/High | 9 | 9 |
| U177 | 2024 | Q5ai | 4 | B×4 | Prove that a continuous p-value is U(0,1) under H0 | K41 | A p-value must be defined or interpreted, or its null distribution proved. | Define the p-value using a prespecified notion of an extreme tail under H0. | Define the tail event → probability-integral transform using a continuous CDF → U(0,1) → P(p<=alpha)=alpha. | Medium/Low/Low | 6 | 9 |
| U178 | 2024 | Q5aii | 3 | A×1 + B×1 + C×1 | Explain the significance threshold and Type I/II errors | K42 | Type I/II errors, significance level and power are at issue. | State the error event separately when H0 is true and when it is false. | Type I=P(reject|H0 true) → Type II=P(fail to reject|alternative) → changing alpha and the power trade-off. | Low/Low/Low | 4 | 9 |
| U179 | 2024 | Q5bi | 2 | A×2 | State H0/H1 for a cola-discrimination experiment | K44 | A fixed randomisation design is given and an exact p-value requested. | Identify the hypergeometric/randomisation statistic under H0 and include outcomes at least as extreme. | Write H0/H1 → exact null distribution → tail sum → compare with alpha. | Low/Low/Low | 3 | 9 |
| U180 | 2024 | Q5bii | 4 | A×2 + B×1 + C×1 | Use exact hypergeometric probabilities to assess whether 5/6 is significant | K10,K44 | A fixed-size sample is drawn without replacement from a population with fixed success/failure counts. | Identify the hypergeometric distribution and specify N, K and n. | Combinatorial numerator/denominator → exact probability or tail probability → check support. Also combine with exact/randomisation tests and hypergeometric tail probabilities. | Medium/Low/Medium | 6 | 9 |
| U181 | 2024 | Q5c | 5 | B×1 + C×1 + D×3 | Prove that X+Y and X-Y are uncorrelated when the variances are equal | K27 | The task concerns correlation, linear-combination correlation or covariance algebra. | Recover covariance from correlation, then calculate the variance of the new variable. | Covariance bilinearity → variance including cross terms → divide by standard deviations → check absolute value <=1. | High/Low/High | 8 | 9 |
| U182 | 2024 | Q5d | 2 | C×2 | Explain why very high sample correlation does not establish causation | K51 | The prompt involves high correlation, aggregation reversal or a causal claim. | Separate sample association from causation and examine trends, confounding and group weights. | Describe correlation → explain why it does not establish causation → confounding/time trends/Simpson mechanism → additional evidence needed. | Low/Low/Low | 3 | 9 |
| U183 | 2024 | Q6a | 3 | B×1 + C×2 | Explain why a 95% confidence interval does not assign 95% probability to the parameter | K38 | A realised confidence interval must be interpreted. | Distinguish a fixed parameter from the random interval before sampling. | Repeated-sampling coverage → fixed realised interval either includes the parameter or does not → reject a posterior-probability interpretation. | Low/Low/Low | 4 | 9 |
| U184 | 2024 | Q6b | 7 | A×3 + B×1 + D×3 | Find the MLE of an exponential distribution's median | K20,K53 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. Also combine with MLEs for regular differentiable likelihoods and log-likelihoods. | High/Low/High | 10 | 9 |
| U185 | 2024 | Q6c | 6 | A×1 + B×4 + D×1 | Find the least-squares coefficient of a quadratic regression model without an intercept | K48 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. | High/Low/High | 9 | 9 |
| U186 | 2024 | Q6d | 2 | B×1 + C×1 | Identify linear-model misspecification from curved residuals | K50 | Residual plots or R² are provided for diagnosis. | Inspect the central residual trend, then check constant variance and systematic patterns. | Identify curvature/funnelling/outliers → relate to model assumptions → propose transformations/extra terms/weights and recheck. | Low/Low/Low | 3 | 9 |
| U187 | 2024 | Q6ei | 1 | A×1 | Reject a proposed bootstrap sample with one observation missing | K52 | A bootstrap sample must be assessed or a bootstrap interval constructed. | Check replacement and equal sample size; for intervals take quantiles of the resampled statistic. | Resample B times → calculate the statistic each time → empirical distribution → 2.5%/97.5% endpoints. | Low/Low/Low | 2 | 9 |
| U188 | 2024 | Q6eii | 1 | A×1 | Recognise an equal-size sample drawn with replacement as a bootstrap sample | K52 | A bootstrap sample must be assessed or a bootstrap interval constructed. | Check replacement and equal sample size; for intervals take quantiles of the resampled statistic. | Resample B times → calculate the statistic each time → empirical distribution → 2.5%/97.5% endpoints. | Low/Low/Low | 2 | 9 |
| U189 | 2025 | Q1ai | 3 | A×1 + B×2 | Describe the probability space for one fair die roll | K01 | A finite, equiprobable or multistage experiment requires Ω, F and P. | Check ordering and replacement assumptions, then write the Cartesian-product sample space. | Sample space → power-set event space → point/event probabilities → check total probability 1. | Low/Low/Low | 4 | 9 |
| U190 | 2025 | Q1aiia | 3 | B×3 | Identify the distributions of individual die results and the match count | K09,K11 | Independent yes/no events are summed into a count. | Define indicators and their success probabilities. | Individual Bernoulli variables → independent sum → Binomial parameters → check the mean. Also combine with discrete-distribution recognition, pmf/CDF and support. | Low/Low/Medium | 4 | 9 |
| U191 | 2025 | Q1aiib | 3 | C×3 | Find the probability of at least one match in the first n trials | K04,K09 | The question compares pairwise/joint independence, complementary events, or distributional conclusions under dependence. | Write the required product equalities, or construct a parity/perfect-dependence counterexample. | Pairwise intersection probabilities → triple intersection/factorisation → determine the level of independence. Also combine with indicators and Bernoulli/Binomial count models. | Low/Low/Medium | 4 | 9 |
| U192 | 2025 | Q1bi | 5 | A×5 | Use total probability with traveller proportions and delay rates to find the probability of lateness | K03 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. | Medium/Low/Low | 8 | 9 |
| U193 | 2025 | Q1bii | 2 | A×2 | Use Bayes' theorem to find the probability of route A given lateness | K08 | Prior rates and sensitivity/false-positive rates are given, and a posterior probability is requested. | Use total probability to form the denominator for the positive-test or observed event. | Define events → total-probability denominator → Bayes inversion → compare with the prior. | Low/Low/Low | 3 | 9 |
| U194 | 2025 | Q1biii | 4 | D×4 | Repeat the Bayes calculation after a signal failure changes conditional probabilities | K03,K08 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. Also combine with Bayes' theorem, diagnostic tests and posterior probabilities. | High/Low/High | 6 | 9 |
| U195 | 2025 | Q2ai | 5 | B×5 | Count passwords containing letters, digits and distinct symbols | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | Medium/Low/Low | 8 | 9 |
| U196 | 2025 | Q2aii | 4 | D×4 | Find the probability that two symbols are adjacent | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | High/Low/High | 6 | 9 |
| U197 | 2025 | Q2bi | 3 | A×3 | Verify a joint density on a triangular region | K17 | A pmf/pdf has an unknown constant, or validity as a density must be assessed. | Determine the support, check nonnegativity, and set the full integral or sum to 1. | Support → nonnegativity → normalising integral/sum → check the constant and unit mass. | Low/Low/Low | 4 | 9 |
| U198 | 2025 | Q2bii | 3 | A×3 | Find the marginal density of X | K18 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. | Low/Low/Low | 4 | 10 |
| U199 | 2025 | Q2biii | 2 | A×2 | Find the conditional density of Y given X | K18 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. | Low/Low/Low | 3 | 10 |
| U200 | 2025 | Q2biv | 3 | C×3 | Use the conditional density to establish dependence and give an alternative test | K19 | The question asks whether continuous variables are independent. | Check both density factorisation and whether the support is a Cartesian product. | Find marginals → compare fXY with fXfY, including zero-density regions → decisive counterexample/conclusion. | Low/Low/Low | 4 | 10 |
| U201 | 2025 | Q3ai | 4 | A×4 | Prove that -log(U)/λ is exponentially distributed | K20,K24 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. Also combine with Normal/Uniform/Exponential/Gamma/Beta distributions. | Medium/Low/Medium | 6 | 10 |
| U202 | 2025 | Q3aii | 4 | A×4 | Find the CDF of the maximum of independent Uniform observations | K21 | The task involves sample maxima, minima or order statistics. | For the maximum CDF, require every observation to be below the threshold; use the survival function for the minimum. | Intersection event → independence product → piecewise CDF → optionally differentiate for a density. | Medium/Low/Low | 6 | 10 |
| U203 | 2025 | Q3bi | 3 | B×3 | Find the MGF of a standard normal variable | K12,K24 | The task involves PGFs/MGFs, identifying a sum's distribution or moment determinacy. | Derive the generating function from its definition; multiply for independent sums. | Defining integral/sum → algebra and domain → compare with standard generating functions → uniqueness. Also combine with Normal/Uniform/Exponential/Gamma/Beta distributions. | Low/Medium/Medium | 4 | 10 |
| U204 | 2025 | Q3bii | 4 | B×1 + C×3 | Use MGFs to identify a sum of independent standard normal variables | K12,K24 | The task involves PGFs/MGFs, identifying a sum's distribution or moment determinacy. | Derive the generating function from its definition; multiply for independent sums. | Defining integral/sum → algebra and domain → compare with standard generating functions → uniqueness. Also combine with Normal/Uniform/Exponential/Gamma/Beta distributions. | Medium/Medium/Medium | 6 | 10 |
| U205 | 2025 | Q3ci | 4 | D×4 | Deduce equality in distribution from equal moments and state additional assumptions | K34 | All moments are said to agree, and equality in distribution is questioned. | Explain why equal moments alone need not suffice; give a sufficient condition such as existence of the MGF near 0. | Moment coefficients → MGF Taylor expansion → equality in a neighbourhood → uniqueness. | High/High/High | 6 | 10 |
| U206 | 2025 | Q3cii | 1 | B×1 | Explain why the conclusion does not depend on continuous/discrete/mixed type | K34 | All moments are said to agree, and equality in distribution is questioned. | Explain why equal moments alone need not suffice; give a sufficient condition such as existence of the MGF near 0. | Moment coefficients → MGF Taylor expansion → equality in a neighbourhood → uniqueness. | Low/Medium/Low | 2 | 10 |
| U207 | 2025 | Q4ai | 2 | A×2 | Find the variance of a sample mean from normal variables with different means and equal variances | K26,K30 | The task requires a mean, variance or covariance identity for linear combinations. | Expand the linear combination using linearity of expectation and bilinearity of covariance. | Linearity of expectation → variance/covariance cross terms → substitute independence or known covariance. Also combine with sample means, sample variances, normal samples and χ² properties. | Low/Low/Medium | 3 | 10 |
| U208 | 2025 | Q4aii | 5 | A×5 | Calculate E[S²] for observations with different means | K26,K30 | The task requires a mean, variance or covariance identity for linear combinations. | Expand the linear combination using linearity of expectation and bilinearity of covariance. | Linearity of expectation → variance/covariance cross terms → substitute independence or known covariance. Also combine with sample means, sample variances, normal samples and χ² properties. | Medium/Low/Medium | 8 | 10 |
| U209 | 2025 | Q4bi | 4 | A×4 | Construct a 90% t interval for a normal mean with unknown variance | K36 | The population is normal with unknown variance and a relatively small sample. | Take the square root of s² to obtain s, and use a t pivot with n-1 degrees of freedom. | Confirm normality and unknown variance → t quantile → standard error → interval. | Medium/Low/Low | 6 | 10 |
| U210 | 2025 | Q4bii | 4 | A×3 + B×1 | Construct a 95% χ² interval for a normal variance | K37 | A confidence interval for a normal population variance is required. | Write (n-1)S²/sigma²~chi-square and note the reversal of endpoint order when inverting. | Chi-square pivot → two tail quantiles → solve for sigma² → verify lower/upper endpoints. | Medium/Low/Low | 6 | 10 |
| U211 | 2025 | Q4c | 3 | B×3 | Interpret a given 99% confidence interval | K38 | A realised confidence interval must be interpreted. | Distinguish a fixed parameter from the random interval before sampling. | Repeated-sampling coverage → fixed realised interval either includes the parameter or does not → reject a posterior-probability interpretation. | Low/Low/Low | 4 | 10 |
| U212 | 2025 | Q4d | 2 | C×2 | Interpret normality evidence from a Q-Q plot | K47 | A scatterplot, boxplot, Q-Q plot or time-series plot must be chosen or interpreted. | Identify the variable type and objective: relationship, distribution, normality or time order. | Choose the graph → identify features to inspect → state a suitably qualified conclusion. | Low/Low/Low | 3 | 10 |
| U213 | 2025 | Q5ai | 1 | A×1 | Calculate the sample median | K46 | Raw data are given and a median, quartiles or outliers are requested. | Sort first and state the quartile convention. | Sort → median and lower/upper halves → IQR → Tukey fences → outliers. | Low/Low/Low | 2 | 10 |
| U214 | 2025 | Q5aii | 2 | A×2 | Calculate the lower and upper quartiles | K46 | Raw data are given and a median, quartiles or outliers are requested. | Sort first and state the quartile convention. | Sort → median and lower/upper halves → IQR → Tukey fences → outliers. | Low/Low/Low | 3 | 10 |
| U215 | 2025 | Q5aiii | 1 | A×1 | Find outliers using Tukey's rule | K46 | Raw data are given and a median, quartiles or outliers are requested. | Sort first and state the quartile convention. | Sort → median and lower/upper halves → IQR → Tukey fences → outliers. | Low/Low/Low | 2 | 10 |
| U216 | 2025 | Q5aiv | 1 | A×1 | Choose a boxplot | K47 | A scatterplot, boxplot, Q-Q plot or time-series plot must be chosen or interpreted. | Identify the variable type and objective: relationship, distribution, normality or time order. | Choose the graph → identify features to inspect → state a suitably qualified conclusion. | Low/Low/Low | 2 | 10 |
| U217 | 2025 | Q5b | 7 | A×1 + B×4 | Perform a two-sample z-test with both variances known and calculate the p-value | K40 | The task tests means or linear combinations from two independent samples. | State independence, normality and known/equal variance assumptions, then write the variances of both sample means. | H0/H1 → pooled-variance or known-variance standard error → statistic/degrees of freedom → p-value. | High/Low/High | 10 | 10 |
| U218 | 2025 | Q5c | 4 | B×3 + D×1 | Define both error types and discuss the effect of changing α | K42 | Type I/II errors, significance level and power are at issue. | State the error event separately when H0 is true and when it is false. | Type I=P(reject|H0 true) → Type II=P(fail to reject|alternative) → changing alpha and the power trade-off. | High/Low/High | 6 | 10 |
| U219 | 2025 | Q5d | 4 | A×1 + B×2 + D×1 | Apply a multiple-comparison correction to ten genetic tests | K41,K43 | A p-value must be defined or interpreted, or its null distribution proved. | Define the p-value using a prespecified notion of an extreme tail under H0. | Define the tail event → probability-integral transform using a continuous CDF → U(0,1) → P(p<=alpha)=alpha. Also combine with multiple testing and Bonferroni correction. | High/Low/High | 6 | 10 |
| U220 | 2025 | Q6a | 6 | A×1 + B×1 + D×4 | Find the MLEs of both endpoints of U[θ1,θ2] | K54 | The parameter determines the support of a Uniform or similar distribution. | Express the requirement that all observations lie in the support as parameter inequalities. | Zero/nonzero likelihood regions → feasible parameters → monotone or flat optimisation → unique/nonunique MLE. | High/Low/High | 9 | 10 |
| U221 | 2025 | Q6bi | 1 | A×1 | Prove an equivalent expression for Sxy | K48 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. | Low/Low/Low | 2 | 11 |
| U222 | 2025 | Q6bii | 1 | A×1 | Express the slope estimator as a linear combination of responses | K48,K49 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. Also combine with variance, covariance and algebraic properties of regression estimators. | Low/Medium/Medium | 2 | 11 |
| U223 | 2025 | Q6biii | 4 | B×1 + C×3 | Express the intercept estimator as a linear combination and find its covariance with the slope | K27,K49 | The task concerns correlation, linear-combination correlation or covariance algebra. | Recover covariance from correlation, then calculate the variance of the new variable. | Covariance bilinearity → variance including cross terms → divide by standard deviations → check absolute value <=1. Also combine with variance, covariance and algebraic properties of regression estimators. | Medium/Medium/Medium | 6 | 11 |
| U224 | 2025 | Q6biv | 4 | C×2 + D×2 | Give a regression example and explain transformations for nonlinearity | K48,K50 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. Also combine with residual diagnostics, limitations of R² and variable transformations. | High/Low/High | 6 | 11 |
| U225 | 2025 | Q6c | 4 | D×4 | Use bootstrap resampling to construct a 95% interval for the class median | K52 | A bootstrap sample must be assessed or a bootstrap interval constructed. | Check replacement and equal sample size; for intervals take quantiles of the resampled statistic. | Resample B times → calculate the statistic each time → empirical distribution → 2.5%/97.5% endpoints. | High/Low/High | 6 | 11 |
| U226 | 2026 | Q1a | 3 | A×3 | Describe the probability space for two consecutive fair six-sided die rolls | K01 | A finite, equiprobable or multistage experiment requires Ω, F and P. | Check ordering and replacement assumptions, then write the Cartesian-product sample space. | Sample space → power-set event space → point/event probabilities → check total probability 1. | Low/Low/Low | 4 | 11 |
| U227 | 2026 | Q1bi | 3 | A×3 | Count all EuroMillions outcomes | K05 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. | Low/Low/Low | 4 | 11 |
| U228 | 2026 | Q1bii | 3 | B×3 | Find the conditional probability of matching both Lucky Stars given exactly 3 main-number matches [source ending truncated] | K03,K10 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. Also combine with finite-population sampling, hypergeometric distributions and exact combinatorial probabilities. | Low/Low/Medium | 4 | 11 |
| U229 | 2026 | Q1biii | 4 | D×4 | Find the overall probability of a secondary prize for a random lottery ticket | K05,K10 | The question involves arrangements, position choices, repeated characters or a double-counting identity. | Count choices of positions separately from choices or arrangements of objects. | Choose positions/objects → correct for repetitions → multiplication principle; for double counting, identify two counts of the same objects. Also combine with finite-population sampling, hypergeometric distributions and exact combinatorial probabilities. | High/Low/High | 6 | 11 |
| U230 | 2026 | Q1biv | 2 | B×2 | Count unordered Lucky Stars multisets when sampling with replacement | K06 | The task concerns ordered positive-integer sums or multiset counting. | Determine whether 0 is allowed; for positive integers, place n-1 separators in the k-1 internal gaps. | Check k>=n → stars and bars → binomial coefficient → check boundary cases. | Low/Low/Low | 3 | 11 |
| U231 | 2026 | Q1c | 5 | A×2 + C×3 | Verify a shifted geometric pmf and find its CDF | K11,K14 | A discrete pmf or geometric-type mass is given, or a CDF on the whole real line is required. | State the support and accumulate mass over its points; use floor for noninteger inputs. | Check nonnegativity and total mass 1 → accumulate piecewise → check right-continuity and limits. Also combine with geometric waiting times, pattern waiting and discrete extremes. | Medium/Low/Medium | 8 | 11 |
| U232 | 2026 | Q2a | 3 | A×3 | Construct three pairwise independent variables that are not jointly independent | K04 | The question compares pairwise/joint independence, complementary events, or distributional conclusions under dependence. | Write the required product equalities, or construct a parity/perfect-dependence counterexample. | Pairwise intersection probabilities → triple intersection/factorisation → determine the level of independence. | Low/Low/Low | 4 | 11 |
| U233 | 2026 | Q2bi | 5 | A×5 | Obtain marginal CDFs/densities from the joint CDF and identify exponential distributions | K18,K24 | Marginal or conditional densities/distributions, or marginals of a joint CDF, are requested. | Fix one variable and determine the integration limits for the other from the support. | Describe support in the other integration order → integrate to obtain marginals → joint/marginal ratio → state conditional support. Also combine with Normal/Uniform/Exponential/Gamma/Beta distributions. | Medium/Low/Medium | 8 | 11 |
| U234 | 2026 | Q2bii | 2 | B×2 | Assess independence by factorising the joint CDF | K19 | The question asks whether continuous variables are independent. | Check both density factorisation and whether the support is a Cartesian product. | Find marginals → compare fXY with fXfY, including zero-density regions → decisive counterexample/conclusion. | Low/Low/Low | 3 | 11 |
| U235 | 2026 | Q2biii | 3 | C×3 | Find the MGF of an exponential variable | K12,K24 | The task involves PGFs/MGFs, identifying a sum's distribution or moment determinacy. | Derive the generating function from its definition; multiply for independent sums. | Defining integral/sum → algebra and domain → compare with standard generating functions → uniqueness. Also combine with Normal/Uniform/Exponential/Gamma/Beta distributions. | Low/Medium/Medium | 4 | 11 |
| U236 | 2026 | Q2biv | 3 | B×3 | Find the CDF of Z=X² | K20 | A continuous variable undergoes a monotone, ratio, product or power transformation. | Find the transformed support and invert the CDF event; use LOTUS directly for expectations. | Support → event inequalities/conditioning → CDF → differentiation/Jacobian → check normalisation. | Low/Low/Low | 4 | 11 |
| U237 | 2026 | Q2ci | 2 | D×2 | Identify the birthday indicators and total count [source task ending is truncated] | K09 | Independent yes/no events are summed into a count. | Define indicators and their success probabilities. | Individual Bernoulli variables → independent sum → Binomial parameters → check the mean. | High/Low/High | 3 | 11 |
| U238 | 2026 | Q2cii | 2 | D×2 | Find the probability that at least one person has a birthday on New Year's Day | K09 | Independent yes/no events are summed into a count. | Define indicators and their success probabilities. | Individual Bernoulli variables → independent sum → Binomial parameters → check the mean. | High/Low/High | 3 | 11 |
| U239 | 2026 | Q3a | 3 | A×2 + B×1 | Identify the squared-loss minimiser and minimum, and the absolute-loss minimiser [source ending truncated] | K29 | The best centre under squared or absolute loss is requested. | For squared loss, add and subtract E[X] and complete the square; for absolute loss, identify a median. | Decompose loss → nonnegative remainder → identify the mean/variance or median. | Low/Low/Low | 4 | 11 |
| U240 | 2026 | Q3b | 3 | A×3 | Prove the bias-variance decomposition of MSE | K31 | An estimator's bias, variance or MSE is requested. | Calculate E and Var separately, then apply MSE=Var+Bias^2. | Expectation → bias → variance → MSE → check nonnegativity and order. | Low/Low/Low | 4 | 11 |
| U241 | 2026 | Q3ci | 1 | A×1 | Choose a scatterplot for the relationship between bedroom count and house price | K47 | A scatterplot, boxplot, Q-Q plot or time-series plot must be chosen or interpreted. | Identify the variable type and objective: relationship, distribution, normality or time order. | Choose the graph → identify features to inspect → state a suitably qualified conclusion. | Low/Low/Low | 2 | 11 |
| U242 | 2026 | Q3cii | 1 | [initial category absent in source] ×1 | Choose a boxplot for house-price median, spread and outliers | K47 | A scatterplot, boxplot, Q-Q plot or time-series plot must be chosen or interpreted. | Identify the variable type and objective: relationship, distribution, normality or time order. | Choose the graph → identify features to inspect → state a suitably qualified conclusion. | Low/Low/Low | 2 | 11 |
| U243 | 2026 | Q3di | 4 | [initial category absent in source] ×4 | Construct a 99% χ² interval for a normal variance | K37 | A confidence interval for a normal population variance is required. | Write (n-1)S²/sigma²~chi-square and note the reversal of endpoint order when inverting. | Chi-square pivot → two tail quantiles → solve for sigma² → verify lower/upper endpoints. | Medium/Low/Low | 6 | 12 |
| U244 | 2026 | Q3dii | 5 | [initial category absent in source] ×3 + C×1 | Perform a two-sided t-test for a normal mean with unknown variance and calculate the p-value | K39 | The task is a one-sample mean hypothesis test. | State H0/H1 and choose z or t according to whether variance is known. | Standardise the statistic → degrees of freedom/tail → p-value/critical region → cautious conclusion. | Medium/Low/Low | 8 | 12 |
| U245 | 2026 | Q3e | 3 | [initial category absent in source] ×2 + C×1 | Define and interpret a p-value and its conditions for a U(0,1) distribution | K41 | A p-value must be defined or interpreted, or its null distribution proved. | Define the p-value using a prespecified notion of an extreme tail under H0. | Define the tail event → probability-integral transform using a continuous CDF → U(0,1) → P(p<=alpha)=alpha. | Low/Low/Low | 4 | 12 |
| U246 | 2026 | Q4a | 4 | [initial category absent in source] ×2 + B×1 | Calculate correlations of linear combinations of correlated normal variables | K27 | The task concerns correlation, linear-combination correlation or covariance algebra. | Recover covariance from correlation, then calculate the variance of the new variable. | Covariance bilinearity → variance including cross terms → divide by standard deviations → check absolute value <=1. | Medium/Low/Low | 6 | 12 |
| U247 | 2026 | Q4b | 3 | [initial category absent in source] ×1 + D×2 | Combine high R² with U-shaped residuals to diagnose misspecification and suggest a transformation | K50 | Residual plots or R² are provided for diagnosis. | Inspect the central residual trend, then check constant variance and systematic patterns. | Identify curvature/funnelling/outliers → relate to model assumptions → propose transformations/extra terms/weights and recheck. | High/Low/High | 4 | 12 |
| U248 | 2026 | Q4c | 2 | [initial category absent in source] ×2 | Give a real-world linear-regression example | K48 | The task requires least-squares derivation, regression coefficients or transformed predictors. | Write RSS and differentiate or complete the square, using the fact that centred values sum to 0. | RSS → normal equations → solve for coefficients → check the minimum and passage through the means. | Low/Low/Low | 3 | 12 |
| U249 | 2026 | Q4d | 2 | [initial category absent in source] ×2 | Explain why highly correlated common trends do not establish causation | K51 | The prompt involves high correlation, aggregation reversal or a causal claim. | Separate sample association from causation and examine trends, confounding and group weights. | Describe correlation → explain why it does not establish causation → confounding/time trends/Simpson mechanism → additional evidence needed. | Low/Low/Low | 3 | 12 |
| U250 | 2026 | Q4ei | 2 | [initial category absent in source] ×2 | Recognise an equal-size sample drawn with replacement as a bootstrap sample | K52 | A bootstrap sample must be assessed or a bootstrap interval constructed. | Check replacement and equal sample size; for intervals take quantiles of the resampled statistic. | Resample B times → calculate the statistic each time → empirical distribution → 2.5%/97.5% endpoints. | Low/Low/Low | 3 | 12 |
| U251 | 2026 | Q4eii | 2 | [initial category absent in source] ×2 | Reject a proposed bootstrap sample of insufficient size | K52 | A bootstrap sample must be assessed or a bootstrap interval constructed. | Check replacement and equal sample size; for intervals take quantiles of the resampled statistic. | Resample B times → calculate the statistic each time → empirical distribution → 2.5%/97.5% endpoints. | Low/Low/Low | 3 | 12 |
| U252 | 2026 | Q4f | 5 | [initial category absent in source] ×2 + D×3 | Find the nonunique MLE under the support constraints of U[θ,θ+1] | K03,K04,K05,K08,K09,K11,K12,K13,K14,K15,K17,K18,K20,K23,K24,K26,K27,K28,K29,K31,K36,K38,K40,K41,K46,K47,K48,K50,K51,K53,K54,K55 | The question gives partition categories and conditional probabilities, or asks for the definition of a probability measure. | For a definition, list the three axioms; for a calculation, first express the categories as a partition. | Nonnegativity/normalisation/countable additivity; or partition → total probability → conditional probability. Also combine with event/random-variable independence, pairwise/joint independence; multiplication, permutations/combinations, repeated elements and positions; Bayes, diagnostics and posteriors; indicators and Bernoulli/Binomial counts; discrete distributions and pmf/CDF/support; generating functions and uniqueness; Poisson superposition and random/compound sums; geometric and pattern waiting and discrete extremes; discrete transformations and mixtures; density validity and normalisation; marginal/conditional distributions; continuous transformations, CDFs, LOTUS and densities; conditional/total expectation and mixtures; Normal/Uniform/Exponential/Gamma/Beta distributions; expectation/variance identities; covariance and correlation; Markov/Chebyshev and bounded variance; squared/absolute-loss minimisers; bias, variance and MSE; Student-t mean intervals; frequentist confidence-interval interpretation; two-sample z/t tests; p-values and U(0,1) under H0; median, quartiles, IQR and Tukey outliers; scatterplots, boxplots, histograms, Q-Q and time-series plots; simple/transformed regression and least squares; residuals and R² limitations; association versus causation, spurious correlation and Simpson's paradox; regular and support-dependent/boundary/nonunique MLEs; Bayesian conjugate updating and posteriors. | High/High/High | 8 | 12 |
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