08 MATH40005 NEW SEVEN YEAR EFFECTIVE COVERAGE MATRIX
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: 7a5d43492c1f5cd862814299ef525c8779f1a27e1ce74ffddc3a69506a4b57baSource date: 2026-08-06
Source page 1
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (8% down the source page)
MATH40005 new effective-coverage matrix across seven years (A+B Definition used )
English passage 2 (14% down the source page)
Under the strict definition ,252 among the ability units A+B Effective coverage 227/252 = 90.1%;5 High-mark abilities at or above this mark level A+B coverage 47/47 = 100.0%.C not counted as effective coverage .
English passage 3 (17% down the source page)
unit
year's question
marks
Task
Revise 1
Revise 2
Revise 3
Revise 4
Revise 5
Revise 6
Best
effective
English passage 4 (18% down the source page)
U001
2020 Q1a
3
definition σ- algebra
D
D
C
A
C
D
A
is
English passage 5 (20% down the source page)
U002
2020 Q1b
3
Define a probability measure
D
C
C
A
D
D
A
is
English passage 6 (22% down the source page)
U003
2020 Q1ci
3
Use total probability to calculate the positive-test rate
D
B
B
C
D
B
B
is
English passage 7 (24% down the source page)
U004
2020 Q1cii
3
use Bayes Calculate disease probability after a positive result
D
B
B
D
D
B
B
is
English passage 8 (25% down the source page)
U005
2020 Q1d
4
Enumerate 7 Express as an ordered sum of three positive integers
C
C
B
B
B
C
B
is
English passage 9 (27% down the source page)
U006
2020 Q1e
4
general k write as n ordered sums of positive integers
D
D
D
A
D
D
A
is
English passage 10 (29% down the source page)
U007
2020 Q2a
3
Prove a given map is a discrete random variable
D
D
C
C
A
D
A
is
English passage 11 (31% down the source page)
U008
2020 Q2b
4
From discrete pmf Write piecewise CDF
C
D
D
A
D
D
A
is
English passage 12 (32% down the source page)
U009
2020 Q2ci
3
Derive the geometric distribution's CDF
A
D
D
B
D
D
A
is
English passage 13 (34% down the source page)
U010
2020 Q2cii
4
Prove that the minimum of two independent geometric variables remains geometric
D
D
C
A
C
D
A
is
English passage 14 (36% down the source page)
U011
2020 Q2d
6
Calculate the first occurrence HT Expected waiting count for a pattern
D
D
C
A
B
C
A
is
English passage 15 (37% down the source page)
U012
2020 Q3a
2
State conditions for a valid probability density function
B
D
C
D
C
B
B
is
English passage 16 (39% down the source page)
U013
2020 Q3bi
2
Assess and rescale 3x is a valid density
B
D
C
D
C
B
B
is
English passage 17 (41% down the source page)
U014
2020 Q3bii
2
Assess a negative constant function and find a scaling constant
B
D
C
D
C
B
B
is
English passage 18 (43% down the source page)
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
English passage 19 (44% down the source page)
U016
2020 Q3ci
3
Normalise a three-dimensional joint density to determine its constant
B
D
B
D
C
B
B
is
English passage 20 (46% down the source page)
U017
2020 Q3cii
3
dimension LOTUS calculation E[XYZ]
B
C
B
D
B
C
B
is
English passage 21 (48% down the source page)
U018
2020 Q3di
3
Partition a sample space and give an example
D
B
B
C
D
D
B
is
English passage 22 (50% down the source page)
U019
2020 Q3dii
3
Discrete law of total expectation
D
C
C
A
C
C
A
is
English passage 23 (51% down the source page)
U020
2020 Q4ai
3
Distribution of a normal sample mean
D
B
B
D
C
C
B
is
English passage 24 (53% down the source page)
U021
2020 Q4aii
1
S Use unbiasedness to find Z Expectation of
C
B
D
C
D
C
B
is
English passage 25 (55% down the source page)
U022
2020 Q4aiii
1
Var(S) find Var(Z)
C
B
D
C
D
C
B
is
English passage 26 (57% down the source page)
U023
2020 Q4aiv
1
Z Estimate σ bias of
C
B
D
A
D
D
A
is
English passage 27 (58% down the source page)
U024
2020 Q4av
2
Z Mean squared error of
C
B
D
A
D
D
A
is
English passage 28 (60% down the source page)
U025
2020 Q4avi
2
Z Obtain χ distribution and degrees of freedom
D
C
D
D
D
C
C
No
English passage 29 (62% down the source page)
U026
2020 Q4avii
2
Use normal-sample independence to find Cov(Xbar,Z)
B
C
D
B
D
C
B
is
English passage 30 (63% down the source page)
U027
2020 Q4b
4
Markov inequality
D
D
D
A
C
D
A
is
English passage 31 (65% down the source page)
U028
2020 Q4c
4
100 Perform for genetic association tests Bonferroni Correction
C
D
D
A
D
C
A
is
English passage 32 (67% down the source page)
U029
2020 Q5a
4
The sample mean minimises the sum of squared deviations
B
D
D
D
D
D
B
is
English passage 33 (69% down the source page)
U030
2020 Q5b
7
Least-squares estimation in simple linear regression
D
C
D
A
C
C
A
is
English passage 34 (70% down the source page)
U031
2020 Q5ci
1
Sample t Null hypothesis of the test
D
B
D
D
D
C
B
is
English passage 35 (72% down the source page)
U032
2020 Q5cii
2
Two-sample t Normality and equal-variance assumptions for the test
D
B
D
D
D
C
B
is
English passage 36 (74% down the source page)
U033
2020 Q5d
6
U[0,θ] Support boundaries of MLE
C
C
D
A
D
C
A
is
English passage 37 (76% down the source page)
U034
2020 Q6ai
2
normal mean with variance 90% Confidence interval
D
D
A
D
D
D
A
is
English passage 38 (77% down the source page)
U035
2020 Q6aii
2
normal mean with variance 95% t Interval
A
B
D
D
D
D
A
is
English passage 39 (79% down the source page)
U036
2020 Q6aiii
2
Chebyshev Construct a distribution-free 99% Mean interval
D
D
D
C
C
D
C
No
English passage 40 (81% down the source page)
U037
2020 Q6bi
1
Joint-distribution assumptions for linear-regression errors
D
C
D
C
C
C
C
No
English passage 41 (82% down the source page)
U038
2020 Q6bii
4
Use two residual plots to assess fit and choose a transformation
C
D
D
D
D
C
C
No
English passage 42 (84% down the source page)
U039
2020 Q6c
3
Likelihood and Gamma Posterior distribution from the prior
D
D
D
A
D
C
A
is
English passage 43 (86% down the source page)
U040
2020 Q6d
6
correlation bounds and Y Boundedness gives sd(X) Lower bound
B
D
D
A
C
D
A
is
English passage 44 (88% down the source page)
U041
2021 Q1a
2
Draw an event in the unit square and assign probability by area
B
D
B
D
B
C
B
is
English passage 45 (89% down the source page)
U042
2021 Q1bi
2
Sample space for repeated die rolls
C
D
D
D
A
C
A
is
Source page 2
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (8% down the source page)
unit
year's question
marks
Task
Revise 1
Revise 2
Revise 3
Revise 4
Revise 5
Revise 6
Best
effective
English passage 2 (10% down the source page)
U043
2021 Q1bii
2
Cardinality of the sample space's power set
C
C
B
C
A
C
A
is
English passage 3 (12% down the source page)
U044
2021 Q1biii
2
Probability space for seven die rolls
C
B
B
C
A
C
A
is
English passage 4 (13% down the source page)
U045
2021 Q1c
2
Complements of independent events remain independent
D
D
C
C
C
D
C
No
English passage 5 (15% down the source page)
U046
2021 Q1di
3
Seven-letter words with distinct consonants and vowels
C
C
B
C
B
C
B
is
English passage 6 (17% down the source page)
U047
2021 Q1dii
3
Seven-letter words with repeated vowels allowed
C
C
B
C
B
C
B
is
English passage 7 (18% down the source page)
U048
2021 Q1e
2
Multiset permutations of race-car colour rankings
C
C
B
C
B
C
B
is
English passage 8 (20% down the source page)
U049
2021 Q1f
2
Use the multiplication principle to count uniform combinations
C
C
B
C
B
C
B
is
English passage 9 (22% down the source page)
U050
2021 Q2a
3
Given the first roll, find the maximum of two rolls' conditional
C
B
B
B
D
D
B
is
English passage 10 (24% down the source page)
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
English passage 11 (25% down the source page)
U052
2021 Q2c
4
Recover marginal distributions by integrating conditional distributions
B
D
C
B
B
B
B
is
English passage 12 (27% down the source page)
U053
2021 Q2d
7
Normal-Gamma of a scale-mixture variable MGF
D
B
B
B
A
C
A
is
English passage 13 (29% down the source page)
U054
2021 Q3ai
2
Uniform of the variable CDF
C
B
B
D
C
C
B
is
English passage 14 (31% down the source page)
U055
2021 Q3aii
3
A shifted variable remains continuous
B
C
C
D
C
B
B
is
English passage 15 (32% down the source page)
U056
2021 Q3aiii
4
translation Uniform Mean and median of
B
B
B
D
C
C
B
is
English passage 16 (34% down the source page)
U057
2021 Q3b
3
LOTUS Usage context and purpose
C
C
C
D
C
C
C
No
English passage 17 (36% down the source page)
U058
2021 Q3ci
1
Normalise the density to determine its constant
B
D
C
D
C
B
B
is
English passage 18 (38% down the source page)
U059
2021 Q3cii
3
Two marginals of a joint density
B
D
C
D
D
B
B
is
English passage 19 (39% down the source page)
U060
2021 Q3ciii
4
P(X/Y≤0.5) Region integral of
A
D
B
D
C
D
A
is
English passage 20 (41% down the source page)
U061
2021 Q4ai
1
Second moment equals squared mean plus variance
D
B
B
D
D
D
B
is
English passage 21 (43% down the source page)
U062
2021 Q4aii
2
Sample second moment is an unbiased estimator
C
B
D
C
C
D
B
is
English passage 22 (44% down the source page)
U063
2021 Q4aiii
3
construct (E[X]) Unbiased estimator of
C
C
D
C
A
D
A
is
English passage 23 (46% down the source page)
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
English passage 24 (48% down the source page)
U065
2021 Q4b
6
use Chebyshev Required sample size for determining average potato weight
D
D
D
B
A
D
A
is
English passage 25 (50% down the source page)
U066
2021 Q4ci
1
Calculate sample median
D
D
D
D
B
D
B
is
English passage 26 (51% down the source page)
U067
2021 Q4cii
2
Calculate lower and upper quartiles
D
D
D
D
B
D
B
is
English passage 27 (53% down the source page)
U068
2021 Q4ciii
2
use Tukey Use the rule to identify outliers
D
D
D
D
B
D
B
is
English passage 28 (55% down the source page)
U069
2021 Q4civ
1
Choose a box plot to show quantiles and outliers
D
C
D
D
D
D
C
No
English passage 29 (57% down the source page)
U070
2021 Q5a
6
for θx^{θ-1} Use the model to find a conventional MLE
D
D
D
D
B
D
B
is
English passage 30 (58% down the source page)
U071
2021 Q5bi
1
Write two-sided hypotheses for an unknown mean
A
D
D
D
D
D
A
is
English passage 31 (60% down the source page)
U072
2021 Q5bii
3
Construct a one-sample t Statistic and critical value
A
D
D
D
D
D
A
is
English passage 32 (62% down the source page)
U073
2021 Q5biii
2
Make a test decision from the statistic
A
D
D
D
D
D
A
is
English passage 33 (63% down the source page)
U074
2021 Q5biv
1
Construct the corresponding 99% t Confidence interval
C
C
D
D
D
D
C
No
English passage 34 (65% down the source page)
U075
2021 Q5c
3
Give a two-sample t Test examples and hypotheses
D
B
D
D
D
C
B
is
English passage 35 (67% down the source page)
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
English passage 36 (69% down the source page)
U077
2021 Q6ai
1
Prove regression-response variance is σ
D
B
D
C
C
C
B
is
English passage 37 (70% down the source page)
U078
2021 Q6aii
2
Rewrite Sxy and express the slope estimator as a linear combination
D
C
D
C
C
C
C
No
English passage 38 (72% down the source page)
U079
2021 Q6aiii
4
calculation Cov(Yi,βhat1)
B
C
D
B
D
D
B
is
English passage 39 (74% down the source page)
U080
2021 Q6aiv
3
Explain a linear-regression example and coefficients' use in Practice Paper
D
C
D
C
C
C
C
No
English passage 40 (76% down the source page)
U081
2021 Q6b
5
Geometric Likelihood and Beta Posterior distribution from the prior
D
D
D
C
D
A
A
is
English passage 41 (77% down the source page)
U082
2021 Q6c
2
From U Detect linear-model misfit from shaped residuals
A
D
D
D
D
C
A
is
English passage 42 (79% down the source page)
U083
2021 Q6di
1
Identify Simpson Paradox
D
D
D
D
C
C
C
No
English passage 43 (81% down the source page)
U084
2021 Q6dii
1
Explain Simpson Conditions producing the paradox
D
D
D
D
C
C
C
No
English passage 44 (83% down the source page)
U085
2021 Q6diii
1
Explain a reversal by stratifying patient risk
D
D
D
D
C
B
B
is
English passage 45 (84% down the source page)
U086
2022 Q1ai
2
Give a nontrivial example on a four-point sample space σ- algebra
D
D
C
C
C
D
C
No
English passage 46 (86% down the source page)
U087
2022 Q1aii
3
Construct a discrete random variable and prove measurability
D
D
C
C
A
D
A
is
English passage 47 (88% down the source page)
U088
2022 Q1aiii
5
Construct a discrete uniform variable and probability measure
B
C
C
C
C
B
B
is
English passage 48 (89% down the source page)
U089
2022 Q1bi
3
Use the binomial theorem / Prove a combinatorial identity by induction
C
C
C
A
C
B
A
is
Source page 3
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (8% down the source page)
unit
year's question
marks
Task
Revise 1
Revise 2
Revise 3
Revise 4
Revise 5
Revise 6
Best
effective
English passage 2 (10% down the source page)
U090
2022 Q1bii
7
provide for a combinatorial identity story proof
C
C
C
A
C
C
A
is
English passage 3 (12% down the source page)
U091
2022 Q2a
3
Prove covariance's bilinear formula for linear combinations
B
D
D
B
D
D
B
is
English passage 4 (13% down the source page)
U092
2022 Q2bi
5
Find expectation and variance of a linear combination
B
B
B
B
D
D
B
is
English passage 5 (15% down the source page)
U093
2022 Q2bii
4
Calculate covariance of two linear combinations
B
D
D
B
D
D
B
is
English passage 6 (17% down the source page)
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
English passage 7 (18% down the source page)
U095
2022 Q2cii
1
Can take 0 when finding Z=XY Mixture of CDF
C
C
C
D
B
C
B
is
English passage 8 (20% down the source page)
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
English passage 9 (22% down the source page)
U097
2022 Q3b
6
Uniform Maximum / Probability of a very rare conditioning event
B
B
B
C
C
C
B
is
English passage 10 (24% down the source page)
U098
2022 Q3ci
1
Normalise the density to determine its constant
B
D
C
D
C
A
A
is
English passage 11 (25% down the source page)
U099
2022 Q3cii
2
X Marginal density of
B
D
C
D
D
A
A
is
English passage 12 (27% down the source page)
U100
2022 Q3ciii
3
E(Y|X=x) and give the domain
A
D
C
C
C
A
A
is
English passage 13 (29% down the source page)
U101
2022 Q4ai
2
Determine estimator bias
C
B
D
A
D
D
A
is
English passage 14 (31% down the source page)
U102
2022 Q4aii
3
of the estimator MSE
C
B
D
A
D
D
A
is
English passage 15 (32% down the source page)
U103
2022 Q4b
6
Bernoulli Variance upper bound and Chebyshev Choose sample size
B
D
D
A
B
D
A
is
English passage 16 (34% down the source page)
U104
2022 Q4c
3
Beta(α,1) Formula for the distribution's median
B
C
C
D
C
C
B
is
English passage 17 (36% down the source page)
U105
2022 Q4di
4
normal mean with variance 99% t Interval
B
B
D
D
D
D
B
is
English passage 18 (38% down the source page)
U106
2022 Q4dii
2
Use a confidence interval to assess whether mean share price is positive
D
D
B
D
D
D
B
is
English passage 19 (39% down the source page)
U107
2022 Q5ai
2
I types and II Type of error
D
D
D
D
B
D
B
is
English passage 20 (41% down the source page)
U108
2022 Q5aii
2
Trade-off between power and both error types
D
D
D
D
B
D
B
is
English passage 21 (43% down the source page)
U109
2022 Q5aiii
2
Explain both error types in a hospital setting
D
D
D
D
B
D
B
is
English passage 22 (44% down the source page)
U110
2022 Q5bi
2
hypotheses for a sided mean test
B
D
D
D
D
D
B
is
English passage 23 (46% down the source page)
U111
2022 Q5bii
4
One-sample with known variance z Test and conclude
B
D
D
D
D
D
B
is
English passage 24 (48% down the source page)
U112
2022 Q5ci
2
Assess the two smallest p-value Whether significant
C
D
D
B
D
C
B
is
English passage 25 (50% down the source page)
U113
2022 Q5cii
2
Adjust the threshold so exactly one discovery is significant
D
D
D
B
D
D
B
is
English passage 26 (51% down the source page)
U114
2022 Q5d
4
Calculate sample correlation from summary statistics
B
D
D
B
D
D
B
is
English passage 27 (53% down the source page)
U115
2022 Q6a
6
of the lognormal location parameter MLE
D
D
D
D
A
D
A
is
English passage 28 (55% down the source page)
U116
2022 Q6bi
1
Determine whether the given sample is bootstrap Sample
A
C
D
D
C
D
A
is
English passage 29 (57% down the source page)
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
English passage 30 (58% down the source page)
U118
2022 Q6c
2
Joint R and residual plots to detect model misfit
C
D
D
D
D
C
C
No
English passage 31 (60% down the source page)
U119
2022 Q6d
4
Use the regression line's required passage through (xbar,ybar) Check coefficients
D
C
D
C
A
C
A
is
English passage 32 (62% down the source page)
U120
2022 Q6e
6
Define likelihood and Gamma Posterior update of the prior
D
D
D
B
D
C
B
is
English passage 33 (63% down the source page)
U121
2023 Q1a
5
0 Probability that at least two people share a birthday weekday
D
D
A
D
D
C
A
is
English passage 34 (65% down the source page)
U122
2023 Q1b
5
Probability that exactly one pair among the people shares a birthday
C
C
A
C
B
A
A
is
English passage 35 (67% down the source page)
U123
2023 Q1c
5
sensitivity / of measles and rash Bayes Posterior
D
B
A
D
D
B
A
is
English passage 36 (69% down the source page)
U124
2023 Q1d
5
Prove that discrete random variables joined by events remain measurable
D
D
A
C
C
D
A
is
English passage 37 (70% down the source page)
U125
2023 Q2ai
2
Prove the indicator and preimage identity
D
D
C
C
A
D
A
is
English passage 38 (72% down the source page)
U126
2023 Q2aii
6
prove σ- The family of preimages of an algebra remains σ- algebra
D
D
C
C
A
D
A
is
English passage 39 (74% down the source page)
U127
2023 Q2b
6
oisson Under random sample size Uniform Minimum tail probability
D
C
A
B
B
C
A
is
English passage 40 (76% down the source page)
U128
2023 Q2c
6
Region probabilities for independent standard normals
B
B
A
D
C
C
A
is
English passage 41 (77% down the source page)
U129
2023 Q3a
3
When independence is absent Poisson The sum need not be Poisson
D
D
A
C
C
C
A
is
English passage 42 (79% down the source page)
U130
2023 Q3bi
1
Normalise the density to determine its constant
A
D
A
D
C
B
A
is
English passage 43 (81% down the source page)
U131
2023 Q3bii
2
marginal densities
A
D
A
D
D
B
A
is
English passage 44 (83% down the source page)
U132
2023 Q3biii
1
X,Y not independent
A
D
A
D
D
D
A
is
English passage 45 (84% down the source page)
U133
2023 Q3biv
4
Y|X=x Conditions of CDF
A
D
A
D
D
B
A
is
English passage 46 (86% down the source page)
U134
2023 Q3ci
3
X^4 of CDF and density
C
B
B
D
C
C
B
is
English passage 47 (88% down the source page)
U135
2023 Q3cii
2
E[X^4]
C
B
B
D
C
C
B
is
English passage 48 (89% down the source page)
U136
2023 Q3d
4
MGF Prove independence Binomial Distribution of the sum
C
C
D
D
C
D
C
No
Source page 4
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (8% down the source page)
unit
year's question
marks
Task
Revise 1
Revise 2
Revise 3
Revise 4
Revise 5
Revise 6
Best
effective
English passage 2 (10% down the source page)
U137
2023 Q4a
3
MSE=Bias+Variance
A
B
D
C
D
D
A
is
English passage 3 (12% down the source page)
U138
2023 Q4b
5
Chebyshev Determine opinion-poll sample size
B
D
D
A
B
D
A
is
English passage 4 (13% down the source page)
U139
2023 Q4ci
4
normal mean with variance 97% Interval
D
D
A
D
D
D
A
is
English passage 5 (15% down the source page)
U140
2023 Q4cii
3
Given 99% Confidence interval
D
D
A
D
D
D
A
is
English passage 6 (17% down the source page)
U141
2023 Q4di
2
Lower and upper quartiles
D
D
D
D
B
D
B
is
English passage 7 (18% down the source page)
U142
2023 Q4dii
3
use Tukey criterion for identifying outliers
D
D
D
D
B
D
B
is
English passage 8 (20% down the source page)
U143
2023 Q5ai
2
definition p-value
C
D
D
D
D
C
C
No
English passage 9 (22% down the source page)
U144
2023 Q5aii
3
Explain p=0.06 Meaning for the experiment
C
D
D
D
D
C
C
No
English passage 10 (24% down the source page)
U145
2023 Q5b
6
Test whether the sum of two normal means equals 360
D
B
D
D
D
A
A
is
English passage 11 (25% down the source page)
U146
2023 Q5c
6
calculation Y and X+3Y Correlation coefficient of
A
D
D
B
D
D
A
is
English passage 12 (27% down the source page)
U147
2023 Q5d
3
Identify and explain in wage data Simpson Paradox
D
D
D
D
A
C
A
is
English passage 13 (29% down the source page)
U148
2023 Q6a
6
Find the geometric-distribution parameter's MLE
D
D
D
D
B
D
B
is
English passage 14 (31% down the source page)
U149
2023 Q6b
6
Derive simple linear regression with an intercept OLS Coefficients
D
C
D
A
C
C
A
is
English passage 15 (32% down the source page)
U150
2023 Q6c
3
Diagnose residual plots and suggest a variable transformation
C
D
D
D
D
C
C
No
English passage 16 (34% down the source page)
U151
2023 Q6di
1
Choose a histogram for a continuous univariate distribution
D
A
D
D
D
D
A
is
English passage 17 (36% down the source page)
U152
2023 Q6dii
1
Choose Q-Q Use a plot to check normality
D
A
D
D
D
D
A
is
English passage 18 (38% down the source page)
U153
2023 Q6diii
1
Choose a bar chart for category proportions / Pie chart
D
A
D
D
D
D
A
is
English passage 19 (39% down the source page)
U154
2023 Q6div
1
Choose a time-series plot for three years of share prices
D
A
D
D
D
D
A
is
English passage 20 (41% down the source page)
U155
2023 Q6dv
1
Choose a box plot for share-price quantiles
D
C
D
D
D
D
C
No
English passage 21 (43% down the source page)
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
English passage 22 (44% down the source page)
U157
2024 Q1bi
4
Random permutations of repeated letters exactly equal BOO
C
C
B
C
B
C
B
is
English passage 23 (46% down the source page)
U158
2024 Q1bii
4
Three symbols in permutations with repeated letters E Probability of adjacency
C
C
B
C
B
C
B
is
English passage 24 (48% down the source page)
U159
2024 Q1c
4
after a positive breast-cancer screening result Bayes Probability
D
B
B
D
D
A
A
is
English passage 25 (50% down the source page)
U160
2024 Q1d
4
Use joint CDF Represent P(X>x,Y>y)
B
B
B
C
D
B
B
is
English passage 26 (51% down the source page)
U161
2024 Q2a
8
of the initial same-face run length of a fair coin PGF
D
C
C
B
B
C
B
is
English passage 27 (53% down the source page)
U162
2024 Q2bi
2
Given a joint density on a triangular region
B
D
C
D
C
B
B
is
English passage 28 (55% down the source page)
U163
2024 Q2bii
2
X Marginal density of
B
D
C
D
D
B
B
is
English passage 29 (57% down the source page)
U164
2024 Q2biii
2
Y Marginal density of
B
D
C
D
D
B
B
is
English passage 30 (58% down the source page)
U165
2024 Q2biv
1
E[X]
C
C
C
D
C
C
C
No
English passage 31 (60% down the source page)
U166
2024 Q2bv
1
E[Y]
C
C
C
D
C
C
C
No
English passage 32 (62% down the source page)
U167
2024 Q2bvi
4
Cov(X,Y)
A
D
D
B
D
D
A
is
English passage 33 (63% down the source page)
U168
2024 Q3a
6
Independent Poisson Conditional on the total, it is Binomial
D
D
B
D
C
C
B
is
English passage 34 (65% down the source page)
U169
2024 Q3bi
4
Poisson First and second moments of the variable
D
B
B
D
B
D
B
is
English passage 35 (67% down the source page)
U170
2024 Q3bii
5
Joint Poisson Expectation of a random sum
D
B
A
C
C
A
A
is
English passage 36 (69% down the source page)
U171
2024 Q3biii
5
Joint Poisson Variance of a random sum
D
B
A
C
C
A
A
is
English passage 37 (70% down the source page)
U172
2024 Q4a
4
Variance is the minimum squared loss
A
D
D
D
D
D
A
is
English passage 38 (72% down the source page)
U173
2024 Q4b
2
Sample median and IQR
D
D
D
D
B
D
B
is
English passage 39 (74% down the source page)
U174
2024 Q4ci
4
normal mean with variance 99% t Interval
A
B
D
D
D
D
A
is
English passage 40 (76% down the source page)
U175
2024 Q4cii
4
Variance 95% χ Interval
D
B
D
D
D
B
B
is
English passage 41 (77% down the source page)
U176
2024 Q4d
6
Uniform Maximum CDF and unbiasedness θ Estimator
C
B
C
C
C
C
B
is
English passage 42 (79% down the source page)
U177
2024 Q5ai
4
prove H0 continuous under p-value Follows U(0,1)
B
D
D
D
D
A
A
is
English passage 43 (81% down the source page)
U178
2024 Q5aii
3
Explain significance thresholds and I/II Type of error
D
D
D
D
B
D
B
is
English passage 44 (83% down the source page)
U179
2024 Q5bi
2
Write for a cola discrimination experiment H0/H1
D
D
B
D
B
D
B
is
English passage 45 (84% down the source page)
U180
2024 Q5bii
4
Assess using exact hypergeometric probability 5/6 Whether significant
D
D
B
D
B
D
B
is
English passage 46 (86% down the source page)
U181
2024 Q5c
5
Prove that with equal variances X+Y and X-Y uncorrelated
A
D
D
B
D
D
A
is
English passage 47 (88% down the source page)
U182
2024 Q5d
2
Explain that extremely high sample correlation does not imply causation
D
D
D
D
C
A
A
is
English passage 48 (89% down the source page)
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
Source page 5
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (8% down the source page)
unit
year's question
marks
Task
Revise 1
Revise 2
Revise 3
Revise 4
Revise 5
Revise 6
Best
effective
English passage 2 (10% down the source page)
U184
2024 Q6b
7
of the median of an exponential distribution MLE
C
C
C
D
B
C
B
is
English passage 3 (12% down the source page)
U185
2024 Q6c
6
Find least-squares coefficients for quadratic regression without an intercept
D
C
D
C
C
A
A
is
English passage 4 (13% down the source page)
U186
2024 Q6d
2
Detect linear-model misfit from a curved residual plot
C
D
D
D
D
C
C
No
English passage 5 (15% down the source page)
U187
2024 Q6ei
1
Determine that a sample one observation short is not bootstrap
A
C
D
D
C
D
A
is
English passage 6 (17% down the source page)
U188
2024 Q6eii
1
Determine that a same-size sample with replacement is bootstrap
A
C
D
D
C
D
A
is
English passage 7 (18% down the source page)
U189
2025 Q1ai
3
Describe the probability space of one fair-die roll
A
D
D
D
B
A
A
is
English passage 8 (20% down the source page)
U190
2025 Q1aiia
3
Identify distributions of each die outcome and the match count
B
D
D
B
D
D
B
is
English passage 9 (22% down the source page)
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
English passage 10 (24% down the source page)
U192
2025 Q1bi
5
Calculate lateness by total probability from passenger proportions and delay rates
D
A
B
C
D
D
A
is
English passage 11 (25% down the source page)
U193
2025 Q1bii
2
Use for lateness Bayes Find the route A Probability of
D
A
B
D
D
B
A
is
English passage 12 (27% down the source page)
U194
2025 Q1biii
4
Recalculate after a fault changes conditional probabilities Bayes
D
A
B
C
D
B
A
is
English passage 13 (29% down the source page)
U195
2025 Q2ai
5
Passwords with letters, digits and distinct symbols
A
A
B
C
B
C
A
is
English passage 14 (31% down the source page)
U196
2025 Q2aii
4
probability of adjacent symbols
A
C
B
C
B
C
A
is
English passage 15 (32% down the source page)
U197
2025 Q2bi
3
Joint density on a triangular region
A
D
C
D
C
B
A
is
English passage 16 (34% down the source page)
U198
2025 Q2bii
3
X Marginal density of
A
D
C
D
D
B
A
is
English passage 17 (36% down the source page)
U199
2025 Q2biii
2
Y|X Conditional density of
A
D
C
D
D
B
A
is
English passage 18 (38% down the source page)
U200
2025 Q2biv
3
Use a conditional density to establish dependence and give an alternative test
A
D
C
D
D
D
A
is
English passage 19 (39% down the source page)
U201
2025 Q3ai
4
-log(U)/λ is exponentially distributed
C
A
C
D
C
C
A
is
English passage 20 (41% down the source page)
U202
2025 Q3aii
4
independent Uniform of the sample maximum CDF
D
A
C
D
D
C
A
is
English passage 21 (43% down the source page)
U203
2025 Q3bi
3
standard normal variable MGF
D
A
B
D
B
C
A
is
English passage 22 (44% down the source page)
U204
2025 Q3bii
4
MGF Find the distribution of a sum of independent standard normals
D
A
B
D
B
C
A
is
English passage 23 (46% down the source page)
U205
2025 Q3ci
4
Equality of all moments implies identical distributions under stated additional assumptions
D
A
D
D
A
D
A
is
English passage 24 (48% down the source page)
U206
2025 Q3cii
1
Conclusion and continuous / Discrete / Independent of mixture type
D
B
D
D
B
D
B
is
English passage 25 (50% down the source page)
U207
2025 Q4ai
2
Variance of the sample mean of normal variables with equal variances
D
A
B
D
D
A
A
is
English passage 26 (51% down the source page)
U208
2025 Q4aii
5
Calculate for a sample with unequal means E[S]
D
A
B
D
D
A
A
is
English passage 27 (53% down the source page)
U209
2025 Q4bi
4
Normal mean with unknown variance 90% t Interval
B
A
D
D
D
D
A
is
English passage 28 (55% down the source page)
U210
2025 Q4bii
4
Normal variance 95% χ Interval
D
A
D
D
D
B
A
is
English passage 29 (57% down the source page)
U211
2025 Q4c
3
Explain the given 99% Confidence interval
D
D
B
D
D
D
B
is
English passage 30 (58% down the source page)
U212
2025 Q4d
2
Explain Q-Q Plot evidence for normality
A
A
D
D
D
D
A
is
English passage 31 (60% down the source page)
U213
2025 Q5ai
1
Calculate sample median
D
D
D
D
B
D
B
is
English passage 32 (62% down the source page)
U214
2025 Q5aii
2
Calculate lower and upper quartiles
D
D
D
D
B
D
B
is
English passage 33 (63% down the source page)
U215
2025 Q5aiii
1
use Tukey Use the rule to find outliers
D
D
D
D
B
D
B
is
English passage 34 (65% down the source page)
U216
2025 Q5aiv
1
Choose a box plot
D
C
D
D
D
D
C
No
English passage 35 (67% down the source page)
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
English passage 36 (69% down the source page)
U218
2025 Q5c
4
Define both error types and discuss α Effect of changes
D
D
D
D
A
D
A
is
English passage 37 (70% down the source page)
U219
2025 Q5d
4
Apply multiple-comparison correction in ten genetic tests
C
D
D
B
D
C
B
is
English passage 38 (72% down the source page)
U220
2025 Q6a
6
U[θ1,θ2] at both endpoints MLE
C
A
D
B
D
C
A
is
English passage 39 (74% down the source page)
U221
2025 Q6bi
1
prove Sxy Equivalent expression for
D
A
D
C
C
C
A
is
English passage 40 (76% down the source page)
U222
2025 Q6bii
1
Express the slope estimator as a linear combination of response variables
D
A
D
C
C
C
A
is
English passage 41 (77% down the source page)
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
English passage 42 (79% down the source page)
U224
2025 Q6biv
4
Give a regression example and explain a transformation for nonlinearity
C
C
D
C
C
C
C
No
English passage 43 (81% down the source page)
U225
2025 Q6c
4
use bootstrap Construct the class median 95% Interval
C
A
D
D
A
D
A
is
English passage 44 (83% down the source page)
U226
2026 Q1a
3
Probability space of two successive fair six-sided die rolls
A
D
D
D
B
C
A
is
English passage 45 (84% down the source page)
U227
2026 Q1bi
3
EuroMillions All outcomes
C
C
B
C
B
C
B
is
English passage 46 (86% down the source page)
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
English passage 47 (88% down the source page)
U229
2026 Q1biii
4
Total probability of secondary prizes in a random lottery
C
C
B
C
B
C
B
is
English passage 48 (89% down the source page)
U230
2026 Q1biv
2
cky Stars Multiset count with replacement and no ordering
D
D
D
B
D
D
B
is
Source page 6
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (8% down the source page)
unit
year's question
marks
Task
Revise 1
Revise 2
Revise 3
Revise 4
Revise 5
Revise 6
Best
effective
English passage 2 (10% down the source page)
U231
2026 Q1c
5
Shifted geometric pmf and find CDF
A
D
D
B
D
D
A
is
English passage 3 (12% down the source page)
U232
2026 Q2a
3
Three variables that are pairwise but not mutually independent
D
D
C
C
A
D
A
is
English passage 4 (13% down the source page)
U233
2026 Q2bi
5
Joint CDF Find the marginal CDF/ density and identify the exponential distribution
B
C
C
D
C
B
B
is
English passage 5 (15% down the source page)
U234
2026 Q2bii
2
Joint CDF Determine independence by factorisation
B
D
C
D
D
D
B
is
English passage 6 (17% down the source page)
U235
2026 Q2biii
3
of a random variable MGF
D
B
B
D
B
C
B
is
English passage 7 (18% down the source page)
U236
2026 Q2biv
3
Z=X of CDF
C
C
B
D
C
C
B
is
English passage 8 (20% down the source page)
U237
2026 Q2ci
2
Identify birthday indicators and the total number of people's
A
D
D
D
D
D
A
is
English passage 9 (22% down the source page)
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
English passage 10 (24% down the source page)
U239
2026 Q3a
3
Optimal point and minimum for squared loss, and optimum absolute loss
A
D
D
D
D
D
A
is
English passage 11 (25% down the source page)
U240
2026 Q3b
3
MSE bias of - Variance decomposition
A
B
D
C
D
D
A
is
English passage 12 (27% down the source page)
U241
2026 Q3ci
1
Choose a scatter plot for bedroom count versus house price
D
A
D
D
D
D
A
is
English passage 13 (29% down the source page)
U242
2026 Q3cii
1
Choose a box plot for median, spread and outliers in house prices
D
A
D
D
D
D
A
is
English passage 14 (31% down the source page)
U243
2026 Q3di
4
normal variance 99% χ Interval
D
B
D
D
D
A
A
is
English passage 15 (32% down the source page)
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
English passage 16 (34% down the source page)
U245
2026 Q3e
3
define and explain p-value and its U(0,1) Condition
A
D
D
D
D
A
A
is
English passage 17 (36% down the source page)
U246
2026 Q4a
4
Calculate correlation of linear combinations of correlated normal variables
A
D
D
B
D
D
A
is
English passage 18 (38% down the source page)
U247
2026 Q4b
3
Joint normal R and U Detect model misfit from shaped residuals and suggest a transformation
A
D
D
D
D
A
A
is
English passage 19 (39% down the source page)
U248
2026 Q4c
2
Give a real-world example of linear regression
D
C
D
C
C
C
C
No
English passage 20 (41% down the source page)
U249
2026 Q4d
2
Explain that common trends with high correlation do not imply causation
D
D
D
D
C
A
A
is
English passage 21 (43% down the source page)
U250
2026 Q4ei
2
Determine that a same-size sample with replacement is bootstrap
A
C
D
D
C
D
A
is
English passage 22 (44% down the source page)
U251
2026 Q4eii
2
Determine that an undersized sample is not bootstrap
A
C
D
D
C
D
A
is
English passage 23 (46% down the source page)
U252
2026 Q4f
5
U[θ,θ+1] Nonunique under support constraints MLE
A
B
B
B
B
A
A
is
Source page 7
