MATH40005 Seven Year Effective Coverage Matrix REVISED 2
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.
Source SHA-256: 8056ff54cce7b487529ae222299e348b929c7c03ddeac2d48a0099d621f33d24Source date: 2026-08-06
EffectiveCoverage
Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.
| unit_id | year | qid | marks | task | R1 | R2 | R3 | R4 | R5 | R6 | new_best | new_true_training | audit_note |
| U001 | 2020 | Q1a | 3 | definition σ- algebra | D | D | C | A | C | D | A | is | |
| U002 | 2020 | Q1b | 3 | Define a probability measure | D | C | C | A | D | D | A | is | |
| U003 | 2020 | Q1ci | 3 | Use total probability to calculate the positive-test rate | D | B | B | C | D | B | B | is | |
| U004 | 2020 | Q1cii | 3 | use Bayes Calculate disease probability after a positive result | D | B | B | D | D | B | B | is | |
| U005 | 2020 | Q1d | 4 | Enumerate 7 Express as an ordered sum of three positive integers | C | C | B | B | B | C | B | is | |
| U006 | 2020 | Q1e | 4 | general k write as n ordered sums of positive integers | D | D | D | A | D | D | A | is | |
| U007 | 2020 | Q2a | 3 | Prove a given map is a discrete random variable | D | D | C | C | A | D | A | is | |
| U008 | 2020 | Q2b | 4 | From discrete pmf Write piecewise CDF | C | D | D | A | D | D | A | is | |
| U009 | 2020 | Q2ci | 3 | Derive the geometric distribution's CDF | A | D | D | B | D | D | A | is | |
| U010 | 2020 | Q2cii | 4 | Prove that the minimum of two independent geometric variables remains geometric | D | D | C | A | C | D | A | is | |
| U011 | 2020 | Q2d | 6 | Calculate the first occurrence HT Expected waiting count for a pattern | D | D | C | A | B | C | A | is | |
| U012 | 2020 | Q3a | 2 | State conditions for a valid probability density function | B | D | C | D | C | B | B | is | |
| U013 | 2020 | Q3bi | 2 | Assess and rescale 3x is a valid density | B | D | C | D | C | B | B | is | |
| U014 | 2020 | Q3bii | 2 | Assess a negative constant function and find a scaling constant | B | D | C | D | C | B | B | is | |
| U015 | 2020 | Q3biii | 2 | Show a piecewise function with positive and negative values cannot be scaled into a density | B | D | C | D | C | B | B | is | |
| U016 | 2020 | Q3ci | 3 | Normalise a three-dimensional joint density to determine its constant | B | D | B | D | C | B | B | is | |
| U017 | 2020 | Q3cii | 3 | dimension LOTUS calculation E[XYZ] | B | C | B | D | B | C | B | is | |
| U018 | 2020 | Q3di | 3 | Partition a sample space and give an example | D | B | B | C | D | D | B | is | |
| U019 | 2020 | Q3dii | 3 | Discrete law of total expectation | D | C | C | A | C | C | A | is | |
| U020 | 2020 | Q4ai | 3 | Distribution of a normal sample mean | D | B | B | D | C | C | B | is | |
| U021 | 2020 | Q4aii | 1 | S² Use unbiasedness to find Z Expectation of | C | B | D | C | D | C | B | is | |
| U022 | 2020 | Q4aiii | 1 | Var(S²) find Var(Z) | C | B | D | C | D | C | B | is | |
| U023 | 2020 | Q4aiv | 1 | Z Estimate σ² bias of | C | B | D | A | D | D | A | is | |
| U024 | 2020 | Q4av | 2 | Z Mean squared error of | C | B | D | A | D | D | A | is | |
| U025 | 2020 | Q4avi | 2 | Z Obtain χ² distribution and degrees of freedom | D | C | D | D | D | C | C | No | |
| U026 | 2020 | Q4avii | 2 | Use normal-sample independence to find Cov(Xbar,Z) | B | C | D | B | D | C | B | is | |
| U027 | 2020 | Q4b | 4 | Markov inequality | D | D | D | A | C | D | A | is | |
| U028 | 2020 | Q4c | 4 | 100 Perform for genetic association tests Bonferroni Correction | C | D | D | A | D | C | A | is | |
| U029 | 2020 | Q5a | 4 | The sample mean minimises the sum of squared deviations | B | D | D | D | D | D | B | is | |
| U030 | 2020 | Q5b | 7 | Least-squares estimation in simple linear regression | D | C | D | A | C | C | A | is | |
| U031 | 2020 | Q5ci | 1 | Sample t Null hypothesis of the test | D | B | D | D | D | C | B | is | |
| U032 | 2020 | Q5cii | 2 | Two-sample t Normality and equal-variance assumptions for the test | D | B | D | D | D | C | B | is | |
| U033 | 2020 | Q5d | 6 | U[0,θ] Support boundaries of MLE | C | C | D | A | D | C | A | is | |
| U034 | 2020 | Q6ai | 2 | normal mean with variance 90% Confidence interval | D | D | A | D | D | D | A | is | |
| U035 | 2020 | Q6aii | 2 | normal mean with variance 95% t Interval | A | B | D | D | D | D | A | is | |
| U036 | 2020 | Q6aiii | 2 | Chebyshev Construct a distribution-free 99% Mean interval | D | D | D | C | C | D | C | No | |
| U037 | 2020 | Q6bi | 1 | Joint-distribution assumptions for linear-regression errors | D | C | D | C | C | C | C | No | |
| U038 | 2020 | Q6bii | 4 | Use two residual plots to assess fit and choose a transformation | C | D | D | D | D | C | C | No | |
| U039 | 2020 | Q6c | 3 | Likelihood and Gamma Posterior distribution from the prior | D | D | D | A | D | C | A | is | |
| U040 | 2020 | Q6d | 6 | correlation bounds and Y Boundedness gives sd(X) Lower bound | B | D | D | A | C | D | A | is | |
| U041 | 2021 | Q1a | 2 | Draw an event in the unit square and assign probability by area | B | D | B | D | B | C | B | is | |
| U042 | 2021 | Q1bi | 2 | Sample space for repeated die rolls | C | D | D | D | A | C | A | is | |
| U043 | 2021 | Q1bii | 2 | Cardinality of the sample space's power set | C | C | B | C | A | C | A | is | |
| U044 | 2021 | Q1biii | 2 | Probability space for seven die rolls | C | B | B | C | A | C | A | is | |
| U045 | 2021 | Q1c | 2 | Complements of independent events remain independent | D | D | C | C | C | D | C | No | |
| U046 | 2021 | Q1di | 3 | Seven-letter words with distinct consonants and vowels | C | C | B | C | B | C | B | is | |
| U047 | 2021 | Q1dii | 3 | Seven-letter words with repeated vowels allowed | C | C | B | C | B | C | B | is | |
| U048 | 2021 | Q1e | 2 | Multiset permutations of race-car colour rankings | C | C | B | C | B | C | B | is | |
| U049 | 2021 | Q1f | 2 | Use the multiplication principle to count uniform combinations | C | C | B | C | B | C | B | is | |
| U050 | 2021 | Q2a | 3 | Given the first roll, find the maximum of two rolls' conditional | C | B | B | B | D | D | B | is | |
| U051 | 2021 | Q2b | 6 | LOTUS and series to find expectation of a function of a geometric variable | C | C | C | B | C | C | B | is | |
| U052 | 2021 | Q2c | 4 | Recover marginal distributions by integrating conditional distributions | B | D | C | B | B | B | B | is | |
| U053 | 2021 | Q2d | 7 | Normal-Gamma of a scale-mixture variable MGF | D | B | B | B | A | C | A | is | |
| U054 | 2021 | Q3ai | 2 | Uniform of the variable CDF | C | B | B | D | C | C | B | is | |
| U055 | 2021 | Q3aii | 3 | A shifted variable remains continuous | B | C | C | D | C | B | B | is | |
| U056 | 2021 | Q3aiii | 4 | translation Uniform Mean and median of | B | B | B | D | C | C | B | is | |
| U057 | 2021 | Q3b | 3 | LOTUS Usage context and purpose | C | C | C | D | C | C | C | No | |
| U058 | 2021 | Q3ci | 1 | Normalise the density to determine its constant | B | D | C | D | C | B | B | is | |
| U059 | 2021 | Q3cii | 3 | Two marginals of a joint density | B | D | C | D | D | B | B | is | |
| U060 | 2021 | Q3ciii | 4 | P(X/Y≤0.5) Region integral of | A | D | B | D | C | D | A | is | |
| U061 | 2021 | Q4ai | 1 | Second moment equals squared mean plus variance | D | B | B | D | D | D | B | is | |
| U062 | 2021 | Q4aii | 2 | Sample second moment is an unbiased estimator | C | B | D | C | C | D | B | is | |
| U063 | 2021 | Q4aiii | 3 | construct (E[X])² Unbiased estimator of | C | C | D | C | A | D | A | is | |
| U064 | 2021 | Q4aiv | 2 | Calculate the estimator's realised value and note the defect that it can be negative | C | B | D | C | C | D | B | is | |
| U065 | 2021 | Q4b | 6 | use Chebyshev Required sample size for determining average potato weight | D | D | D | B | A | D | A | is | AUDITED_REVISED_2: R5-3(d) numerical endpoint corrected to n=2501; A-level direct coverage unchanged. |
| U066 | 2021 | Q4ci | 1 | Calculate sample median | D | D | D | D | B | D | B | is | |
| U067 | 2021 | Q4cii | 2 | Calculate lower and upper quartiles | D | D | D | D | B | D | B | is | |
| U068 | 2021 | Q4ciii | 2 | use Tukey Use the rule to identify outliers | D | D | D | D | B | D | B | is | |
| U069 | 2021 | Q4civ | 1 | Choose a box plot to show quantiles and outliers | D | C | D | D | D | D | C | No | |
| U070 | 2021 | Q5a | 6 | for θx^{θ-1} Use the model to find a conventional MLE | D | D | D | D | B | D | B | is | |
| U071 | 2021 | Q5bi | 1 | Write two-sided hypotheses for an unknown mean | A | D | D | D | D | D | A | is | |
| U072 | 2021 | Q5bii | 3 | Construct a one-sample t Statistic and critical value | A | D | D | D | D | D | A | is | |
| U073 | 2021 | Q5biii | 2 | Make a test decision from the statistic | A | D | D | D | D | D | A | is | |
| U074 | 2021 | Q5biv | 1 | Construct the corresponding 99% t Confidence interval | C | C | D | D | D | D | C | No | |
| U075 | 2021 | Q5c | 3 | Give a two-sample t Test examples and hypotheses | D | B | D | D | D | C | B | is | |
| U076 | 2021 | Q5d | 4 | H0 when all are true 200 items p-value below 0.05 Expected count of | A | D | D | D | D | C | A | is | |
| U077 | 2021 | Q6ai | 1 | Prove regression-response variance is σ² | D | B | D | C | C | C | B | is | |
| U078 | 2021 | Q6aii | 2 | Rewrite Sxy and express the slope estimator as a linear combination | D | C | D | C | C | C | C | No | |
| U079 | 2021 | Q6aiii | 4 | calculation Cov(Yi,βhat1) | B | C | D | B | D | D | B | is | |
| U080 | 2021 | Q6aiv | 3 | Explain a linear-regression example and coefficients' use in Practice Paper | D | C | D | C | C | C | C | No | |
| U081 | 2021 | Q6b | 5 | Geometric Likelihood and Beta Posterior distribution from the prior | D | D | D | C | D | A | A | is | |
| U082 | 2021 | Q6c | 2 | From U Detect linear-model misfit from shaped residuals | A | D | D | D | D | C | A | is | |
| U083 | 2021 | Q6di | 1 | Identify Simpson Paradox | D | D | D | D | C | C | C | No | |
| U084 | 2021 | Q6dii | 1 | Explain Simpson Conditions producing the paradox | D | D | D | D | C | C | C | No | |
| U085 | 2021 | Q6diii | 1 | Explain a reversal by stratifying patient risk | D | D | D | D | C | B | B | is | |
| U086 | 2022 | Q1ai | 2 | Give a nontrivial example on a four-point sample space σ- algebra | D | D | C | C | C | D | C | No | |
| U087 | 2022 | Q1aii | 3 | Construct a discrete random variable and prove measurability | D | D | C | C | A | D | A | is | |
| U088 | 2022 | Q1aiii | 5 | Construct a discrete uniform variable and probability measure | B | C | C | C | C | B | B | is | |
| U089 | 2022 | Q1bi | 3 | Use the binomial theorem / Prove a combinatorial identity by induction | C | C | C | A | C | B | A | is | |
| U090 | 2022 | Q1bii | 7 | provide for a combinatorial identity story proof | C | C | C | A | C | C | A | is | |
| U091 | 2022 | Q2a | 3 | Prove covariance's bilinear formula for linear combinations | B | D | D | B | D | D | B | is | |
| U092 | 2022 | Q2bi | 5 | Find expectation and variance of a linear combination | B | B | B | B | D | D | B | is | |
| U093 | 2022 | Q2bii | 4 | Calculate covariance of two linear combinations | B | D | D | B | D | D | B | is | |
| U094 | 2022 | Q2ci | 7 | X Continuous , Y For discrete variables find Z=XY density of and verify | C | C | C | C | A | C | A | is | |
| U095 | 2022 | Q2cii | 1 | Can take 0 when finding Z=XY Mixture of CDF | C | C | C | D | B | C | B | is | |
| U096 | 2022 | Q3a | 8 | Low values reported in the media / with high disease prevalence Bayes Conclude and identify | D | B | B | D | D | A | A | is | |
| U097 | 2022 | Q3b | 6 | Uniform Maximum / Probability of a very rare conditioning event | B | B | B | C | C | C | B | is | |
| U098 | 2022 | Q3ci | 1 | Normalise the density to determine its constant | B | D | C | D | C | A | A | is | |
| U099 | 2022 | Q3cii | 2 | X Marginal density of | B | D | C | D | D | A | A | is | |
| U100 | 2022 | Q3ciii | 3 | E(Y|X=x) and give the domain | A | D | C | C | C | A | A | is | |
| U101 | 2022 | Q4ai | 2 | Determine estimator bias | C | B | D | A | D | D | A | is | |
| U102 | 2022 | Q4aii | 3 | of the estimator MSE | C | B | D | A | D | D | A | is | |
| U103 | 2022 | Q4b | 6 | Bernoulli Variance upper bound and Chebyshev Choose sample size | B | D | D | A | B | D | A | is | |
| U104 | 2022 | Q4c | 3 | Beta(α,1) Formula for the distribution's median | B | C | C | D | C | C | B | is | |
| U105 | 2022 | Q4di | 4 | normal mean with variance 99% t Interval | B | B | D | D | D | D | B | is | |
| U106 | 2022 | Q4dii | 2 | Use a confidence interval to assess whether mean share price is positive | D | D | B | D | D | D | B | is | |
| U107 | 2022 | Q5ai | 2 | I types and II Type of error | D | D | D | D | B | D | B | is | |
| U108 | 2022 | Q5aii | 2 | Trade-off between power and both error types | D | D | D | D | B | D | B | is | |
| U109 | 2022 | Q5aiii | 2 | Explain both error types in a hospital setting | D | D | D | D | B | D | B | is | |
| U110 | 2022 | Q5bi | 2 | hypotheses for a sided mean test | B | D | D | D | D | D | B | is | |
| U111 | 2022 | Q5bii | 4 | One-sample with known variance z Test and conclude | B | D | D | D | D | D | B | is | |
| U112 | 2022 | Q5ci | 2 | Assess the two smallest p-value Whether significant | C | D | D | B | D | C | B | is | |
| U113 | 2022 | Q5cii | 2 | Adjust the threshold so exactly one discovery is significant | D | D | D | B | D | D | B | is | |
| U114 | 2022 | Q5d | 4 | Calculate sample correlation from summary statistics | B | D | D | B | D | D | B | is | |
| U115 | 2022 | Q6a | 6 | of the lognormal location parameter MLE | D | D | D | D | A | D | A | is | |
| U116 | 2022 | Q6bi | 1 | Determine whether the given sample is bootstrap Sample | A | C | D | D | C | D | A | is | |
| U117 | 2022 | Q6bii | 1 | Determine that a sample containing values outside the original sample is not bootstrap | A | C | D | D | C | D | A | is | |
| U118 | 2022 | Q6c | 2 | Joint R² and residual plots to detect model misfit | C | D | D | D | D | C | C | No | |
| U119 | 2022 | Q6d | 4 | Use the regression line's required passage through (xbar,ybar) Check coefficients | D | C | D | C | A | C | A | is | |
| U120 | 2022 | Q6e | 6 | Define likelihood and Gamma Posterior update of the prior | D | D | D | B | D | C | B | is | |
| U121 | 2023 | Q1a | 5 | 0 Probability that at least two people share a birthday weekday | D | D | A | D | D | C | A | is | |
| U122 | 2023 | Q1b | 5 | Probability that exactly one pair among the people shares a birthday | C | C | A | C | B | A | A | is | |
| U123 | 2023 | Q1c | 5 | sensitivity / of measles and rash Bayes Posterior | D | B | A | D | D | B | A | is | |
| U124 | 2023 | Q1d | 5 | Prove that discrete random variables joined by events remain measurable | D | D | A | C | C | D | A | is | |
| U125 | 2023 | Q2ai | 2 | Prove the indicator and preimage identity | D | D | C | C | A | D | A | is | |
| U126 | 2023 | Q2aii | 6 | prove σ- The family of preimages of an algebra remains σ- algebra | D | D | C | C | A | D | A | is | |
| U127 | 2023 | Q2b | 6 | oisson Under random sample size Uniform Minimum tail probability | D | C | A | B | B | C | A | is | |
| U128 | 2023 | Q2c | 6 | Region probabilities for independent standard normals | B | B | A | D | C | C | A | is | |
| U129 | 2023 | Q3a | 3 | When independence is absent Poisson The sum need not be Poisson | D | D | A | C | C | C | A | is | |
| U130 | 2023 | Q3bi | 1 | Normalise the density to determine its constant | A | D | A | D | C | B | A | is | |
| U131 | 2023 | Q3bii | 2 | marginal densities | A | D | A | D | D | B | A | is | |
| U132 | 2023 | Q3biii | 1 | X,Y not independent | A | D | A | D | D | D | A | is | |
| U133 | 2023 | Q3biv | 4 | Y|X=x Conditions of CDF | A | D | A | D | D | B | A | is | |
| U134 | 2023 | Q3ci | 3 | X^4 of CDF and density | C | B | B | D | C | C | B | is | |
| U135 | 2023 | Q3cii | 2 | E[X^4] | C | B | B | D | C | C | B | is | |
| U136 | 2023 | Q3d | 4 | MGF Prove independence Binomial Distribution of the sum | C | C | D | D | C | D | C | No | |
| U137 | 2023 | Q4a | 3 | MSE=Bias²+Variance | A | B | D | C | D | D | A | is | |
| U138 | 2023 | Q4b | 5 | Chebyshev Determine opinion-poll sample size | B | D | D | A | B | D | A | is | |
| U139 | 2023 | Q4ci | 4 | normal mean with variance 97% Interval | D | D | A | D | D | D | A | is | |
| U140 | 2023 | Q4cii | 3 | Given 99% Confidence interval | D | D | A | D | D | D | A | is | |
| U141 | 2023 | Q4di | 2 | Lower and upper quartiles | D | D | D | D | B | D | B | is | |
| U142 | 2023 | Q4dii | 3 | use Tukey criterion for identifying outliers | D | D | D | D | B | D | B | is | |
| U143 | 2023 | Q5ai | 2 | definition p-value | C | D | D | D | D | C | C | No | |
| U144 | 2023 | Q5aii | 3 | Explain p=0.06 Meaning for the experiment | C | D | D | D | D | C | C | No | |
| U145 | 2023 | Q5b | 6 | Test whether the sum of two normal means equals 360 | D | B | D | D | D | A | A | is | |
| U146 | 2023 | Q5c | 6 | calculation Y and X+3Y Correlation coefficient of | A | D | D | B | D | D | A | is | |
| U147 | 2023 | Q5d | 3 | Identify and explain in wage data Simpson Paradox | D | D | D | D | A | C | A | is | |
| U148 | 2023 | Q6a | 6 | Find the geometric-distribution parameter's MLE | D | D | D | D | B | D | B | is | |
| U149 | 2023 | Q6b | 6 | Derive simple linear regression with an intercept OLS Coefficients | D | C | D | A | C | C | A | is | |
| U150 | 2023 | Q6c | 3 | Diagnose residual plots and suggest a variable transformation | C | D | D | D | D | C | C | No | |
| U151 | 2023 | Q6di | 1 | Choose a histogram for a continuous univariate distribution | D | A | D | D | D | D | A | is | |
| U152 | 2023 | Q6dii | 1 | Choose Q-Q Use a plot to check normality | D | A | D | D | D | D | A | is | |
| U153 | 2023 | Q6diii | 1 | Choose a bar chart for category proportions / Pie chart | D | A | D | D | D | D | A | is | |
| U154 | 2023 | Q6div | 1 | Choose a time-series plot for three years of share prices | D | A | D | D | D | D | A | is | |
| U155 | 2023 | Q6dv | 1 | Choose a box plot for share-price quantiles | D | C | D | D | D | D | C | No | |
| U156 | 2024 | Q1a | 4 | 50 identical footballs distributed to 20 people, each receiving at least 1 items | D | D | D | B | D | D | B | is | |
| U157 | 2024 | Q1bi | 4 | Random permutations of repeated letters exactly equal BOO | C | C | B | C | B | C | B | is | |
| U158 | 2024 | Q1bii | 4 | Three symbols in permutations with repeated letters E Probability of adjacency | C | C | B | C | B | C | B | is | |
| U159 | 2024 | Q1c | 4 | after a positive breast-cancer screening result Bayes Probability | D | B | B | D | D | A | A | is | |
| U160 | 2024 | Q1d | 4 | Use joint CDF Represent P(X>x,Y>y) | B | B | B | C | D | B | B | is | |
| U161 | 2024 | Q2a | 8 | of the initial same-face run length of a fair coin PGF | D | C | C | B | B | C | B | is | |
| U162 | 2024 | Q2bi | 2 | Given a joint density on a triangular region | B | D | C | D | C | B | B | is | |
| U163 | 2024 | Q2bii | 2 | X Marginal density of | B | D | C | D | D | B | B | is | |
| U164 | 2024 | Q2biii | 2 | Y Marginal density of | B | D | C | D | D | B | B | is | |
| U165 | 2024 | Q2biv | 1 | E[X] | C | C | C | D | C | C | C | No | |
| U166 | 2024 | Q2bv | 1 | E[Y] | C | C | C | D | C | C | C | No | |
| U167 | 2024 | Q2bvi | 4 | Cov(X,Y) | A | D | D | B | D | D | A | is | |
| U168 | 2024 | Q3a | 6 | Independent Poisson Conditional on the total, it is Binomial | D | D | B | D | C | C | B | is | |
| U169 | 2024 | Q3bi | 4 | Poisson First and second moments of the variable | D | B | B | D | B | D | B | is | |
| U170 | 2024 | Q3bii | 5 | Joint Poisson Expectation of a random sum | D | B | A | C | C | A | A | is | |
| U171 | 2024 | Q3biii | 5 | Joint Poisson Variance of a random sum | D | B | A | C | C | A | A | is | |
| U172 | 2024 | Q4a | 4 | Variance is the minimum squared loss | A | D | D | D | D | D | A | is | |
| U173 | 2024 | Q4b | 2 | Sample median and IQR | D | D | D | D | B | D | B | is | |
| U174 | 2024 | Q4ci | 4 | normal mean with variance 99% t Interval | A | B | D | D | D | D | A | is | |
| U175 | 2024 | Q4cii | 4 | Variance 95% χ² Interval | D | B | D | D | D | B | B | is | |
| U176 | 2024 | Q4d | 6 | Uniform Maximum CDF and unbiasedness θ Estimator | C | B | C | C | C | C | B | is | |
| U177 | 2024 | Q5ai | 4 | prove H0 continuous under p-value Follows U(0,1) | B | D | D | D | D | A | A | is | |
| U178 | 2024 | Q5aii | 3 | Explain significance thresholds and I/II Type of error | D | D | D | D | B | D | B | is | |
| U179 | 2024 | Q5bi | 2 | Write for a cola discrimination experiment H0/H1 | D | D | B | D | B | D | B | is | |
| U180 | 2024 | Q5bii | 4 | Assess using exact hypergeometric probability 5/6 Whether significant | D | D | B | D | B | D | B | is | |
| U181 | 2024 | Q5c | 5 | Prove that with equal variances X+Y and X-Y uncorrelated | A | D | D | B | D | D | A | is | |
| U182 | 2024 | Q5d | 2 | Explain that extremely high sample correlation does not imply causation | D | D | D | D | C | A | A | is | |
| U183 | 2024 | Q6a | 3 | Explain 95% A confidence interval does not assign the parameter a 95% Probability | D | D | A | D | D | D | A | is | |
| U184 | 2024 | Q6b | 7 | of the median of an exponential distribution MLE | C | C | C | D | B | C | B | is | |
| U185 | 2024 | Q6c | 6 | Find least-squares coefficients for quadratic regression without an intercept | D | C | D | C | C | A | A | is | |
| U186 | 2024 | Q6d | 2 | Detect linear-model misfit from a curved residual plot | C | D | D | D | D | C | C | No | |
| U187 | 2024 | Q6ei | 1 | Determine that a sample one observation short is not bootstrap | A | C | D | D | C | D | A | is | |
| U188 | 2024 | Q6eii | 1 | Determine that a same-size sample with replacement is bootstrap | A | C | D | D | C | D | A | is | |
| U189 | 2025 | Q1ai | 3 | Describe the probability space of one fair-die roll | A | D | D | D | B | A | A | is | |
| U190 | 2025 | Q1aiia | 3 | Identify distributions of each die outcome and the match count | B | D | D | B | D | D | B | is | |
| U191 | 2025 | Q1aiib | 3 | Find the first n Probability of at least one match in repeated trials | B | D | B | C | C | D | B | is | |
| U192 | 2025 | Q1bi | 5 | Calculate lateness by total probability from passenger proportions and delay rates | D | A | B | C | D | D | A | is | |
| U193 | 2025 | Q1bii | 2 | Use for lateness Bayes Find the route A Probability of | D | A | B | D | D | B | A | is | |
| U194 | 2025 | Q1biii | 4 | Recalculate after a fault changes conditional probabilities Bayes | D | A | B | C | D | B | A | is | |
| U195 | 2025 | Q2ai | 5 | Passwords with letters, digits and distinct symbols | A | A | B | C | B | C | A | is | |
| U196 | 2025 | Q2aii | 4 | probability of adjacent symbols | A | C | B | C | B | C | A | is | |
| U197 | 2025 | Q2bi | 3 | Joint density on a triangular region | A | D | C | D | C | B | A | is | |
| U198 | 2025 | Q2bii | 3 | X Marginal density of | A | D | C | D | D | B | A | is | |
| U199 | 2025 | Q2biii | 2 | Y|X Conditional density of | A | D | C | D | D | B | A | is | |
| U200 | 2025 | Q2biv | 3 | Use a conditional density to establish dependence and give an alternative test | A | D | C | D | D | D | A | is | |
| U201 | 2025 | Q3ai | 4 | -log(U)/λ is exponentially distributed | C | A | C | D | C | C | A | is | |
| U202 | 2025 | Q3aii | 4 | independent Uniform of the sample maximum CDF | D | A | C | D | D | C | A | is | |
| U203 | 2025 | Q3bi | 3 | standard normal variable MGF | D | A | B | D | B | C | A | is | |
| U204 | 2025 | Q3bii | 4 | MGF Find the distribution of a sum of independent standard normals | D | A | B | D | B | C | A | is | |
| U205 | 2025 | Q3ci | 4 | Equality of all moments implies identical distributions under stated additional assumptions | D | A | D | D | A | D | A | is | |
| U206 | 2025 | Q3cii | 1 | Conclusion and continuous / Discrete / Independent of mixture type | D | B | D | D | B | D | B | is | |
| U207 | 2025 | Q4ai | 2 | Variance of the sample mean of normal variables with equal variances | D | A | B | D | D | A | A | is | |
| U208 | 2025 | Q4aii | 5 | Calculate for a sample with unequal means E[S²] | D | A | B | D | D | A | A | is | |
| U209 | 2025 | Q4bi | 4 | Normal mean with unknown variance 90% t Interval | B | A | D | D | D | D | A | is | |
| U210 | 2025 | Q4bii | 4 | Normal variance 95% χ² Interval | D | A | D | D | D | B | A | is | |
| U211 | 2025 | Q4c | 3 | Explain the given 99% Confidence interval | D | D | B | D | D | D | B | is | |
| U212 | 2025 | Q4d | 2 | Explain Q-Q Plot evidence for normality | A | A | D | D | D | D | A | is | |
| U213 | 2025 | Q5ai | 1 | Calculate sample median | D | D | D | D | B | D | B | is | |
| U214 | 2025 | Q5aii | 2 | Calculate lower and upper quartiles | D | D | D | D | B | D | B | is | |
| U215 | 2025 | Q5aiii | 1 | use Tukey Use the rule to find outliers | D | D | D | D | B | D | B | is | |
| U216 | 2025 | Q5aiv | 1 | Choose a box plot | D | C | D | D | D | D | C | No | |
| U217 | 2025 | Q5b | 7 | Two-sample procedure with both variances known z Test and calculate p-value | D | A | D | D | D | C | A | is | |
| U218 | 2025 | Q5c | 4 | Define both error types and discuss α Effect of changes | D | D | D | D | A | D | A | is | |
| U219 | 2025 | Q5d | 4 | Apply multiple-comparison correction in ten genetic tests | C | D | D | B | D | C | B | is | |
| U220 | 2025 | Q6a | 6 | U[θ1,θ2] at both endpoints MLE | C | A | D | B | D | C | A | is | |
| U221 | 2025 | Q6bi | 1 | prove Sxy Equivalent expression for | D | A | D | C | C | C | A | is | |
| U222 | 2025 | Q6bii | 1 | Express the slope estimator as a linear combination of response variables | D | A | D | C | C | C | A | is | |
| U223 | 2025 | Q6biii | 4 | Express the intercept estimator as a linear combination and find covariance with the slope | B | A | D | B | D | D | A | is | |
| U224 | 2025 | Q6biv | 4 | Give a regression example and explain a transformation for nonlinearity | C | C | D | C | C | C | C | No | |
| U225 | 2025 | Q6c | 4 | use bootstrap Construct the class median 95% Interval | C | A | D | D | A | D | A | is | |
| U226 | 2026 | Q1a | 3 | Probability space of two successive fair six-sided die rolls | A | D | D | D | B | C | A | is | |
| U227 | 2026 | Q1bi | 3 | EuroMillions All outcomes | C | C | B | C | B | C | B | is | |
| U228 | 2026 | Q1bii | 3 | Exactly match 3 matching two of the main numbers Lucky Stars conditions of | D | B | B | C | C | D | B | is | |
| U229 | 2026 | Q1biii | 4 | Total probability of secondary prizes in a random lottery | C | C | B | C | B | C | B | is | |
| U230 | 2026 | Q1biv | 2 | cky Stars Multiset count with replacement and no ordering | D | D | D | B | D | D | B | is | |
| U231 | 2026 | Q1c | 5 | Shifted geometric pmf and find CDF | A | D | D | B | D | D | A | is | |
| U232 | 2026 | Q2a | 3 | Three variables that are pairwise but not mutually independent | D | D | C | C | A | D | A | is | |
| U233 | 2026 | Q2bi | 5 | Joint CDF Find the marginal CDF/ density and identify the exponential distribution | B | C | C | D | C | B | B | is | |
| U234 | 2026 | Q2bii | 2 | Joint CDF Determine independence by factorisation | B | D | C | D | D | D | B | is | |
| U235 | 2026 | Q2biii | 3 | of a random variable MGF | D | B | B | D | B | C | B | is | |
| U236 | 2026 | Q2biv | 3 | Z=X² of CDF | C | C | B | D | C | C | B | is | |
| U237 | 2026 | Q2ci | 2 | Identify birthday indicators and the total number of people's | A | D | D | D | D | D | A | is | |
| U238 | 2026 | Q2cii | 2 | Probability that at least one person has a New Year's Day birthday | B | D | D | D | D | D | B | is | |
| U239 | 2026 | Q3a | 3 | Optimal point and minimum for squared loss, and optimum absolute loss | A | D | D | D | D | D | A | is | |
| U240 | 2026 | Q3b | 3 | MSE bias of - Variance decomposition | A | B | D | C | D | D | A | is | |
| U241 | 2026 | Q3ci | 1 | Choose a scatter plot for bedroom count versus house price | D | A | D | D | D | D | A | is | |
| U242 | 2026 | Q3cii | 1 | Choose a box plot for median, spread and outliers in house prices | D | A | D | D | D | D | A | is | |
| U243 | 2026 | Q3di | 4 | normal variance 99% χ² Interval | D | B | D | D | D | A | A | is | |
| U244 | 2026 | Q3dii | 5 | Two-sided normal mean with known variance t Test and calculate p-value | A | D | D | D | D | D | A | is | |
| U245 | 2026 | Q3e | 3 | define and explain p-value and its U(0,1) Condition | A | D | D | D | D | A | A | is | |
| U246 | 2026 | Q4a | 4 | Calculate correlation of linear combinations of correlated normal variables | A | D | D | B | D | D | A | is | |
| U247 | 2026 | Q4b | 3 | Joint normal R² and U Detect model misfit from shaped residuals and suggest a transformation | A | D | D | D | D | A | A | is | |
| U248 | 2026 | Q4c | 2 | Give a real-world example of linear regression | D | C | D | C | C | C | C | No | |
| U249 | 2026 | Q4d | 2 | Explain that common trends with high correlation do not imply causation | D | D | D | D | C | A | A | is | |
| U250 | 2026 | Q4ei | 2 | Determine that a same-size sample with replacement is bootstrap | A | C | D | D | C | D | A | is | |
| U251 | 2026 | Q4eii | 2 | Determine that an undersized sample is not bootstrap | A | C | D | D | C | D | A | is | |
| U252 | 2026 | Q4f | 5 | U[θ,θ+1] Nonunique under support constraints MLE | A | B | B | B | B | A | A | is |