07 MATH40005 SIX PAPER DIFFICULTY CALIBRATION

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: dbae19dcbd1a4ad1d56a755f2c5315cfba090e0035de14bcb95228c022e17313
Source 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 Difficulty calibration report for six revised papers

English passage 2 (14% down the source page) Calibration benchmark

English passage 3 (18% down the source page) The 2026 official paper is 2 hours, 4 questions and 80 marks. Its solutions’ final page gives A=32, B=20, C=12 and D=16. All subpart allocations in these six papers reconcile to the same total difficulty budget, without fixed copying of 2026 question types, question numbers or method sequences.

English passage 5 (23% down the source page) 32 20 12 16 4 questions /80 marks /120 minutes Approximately per question 3-8 substantive steps; numerical questions are solvable by hand / Look up a table Each paper contains definitions / prove / modelling / a mixture of explanations, rather than primarily advanced theory Allocated time totals approximately 117 minutes; reserve additional checking time

English passage 6 (26% down the source page) 32 20 12 16 4 questions /80 marks /120 minutes Approximately per question 3-8 substantive steps; numerical questions are solvable by hand / Look up a table Each paper contains definitions / prove / modelling / a mixture of explanations, rather than primarily advanced theory Allocated time totals approximately 109 minutes; reserve additional checking time

English passage 7 (29% down the source page) 32 20 12 16 4 questions /80 marks /120 minutes Approximately per question 3-8 substantive steps; numerical questions are solvable by hand / Look up a table Each paper contains definitions / prove / modelling / a mixture of explanations, rather than primarily advanced theory Allocated time totals approximately 109 minutes; reserve additional checking time

English passage 8 (33% down the source page) 32 20 12 16 4 questions /80 marks /120 minutes Approximately per question 3-8 substantive steps; numerical questions are solvable by hand / Look up a table Each paper contains definitions / prove / modelling / a mixture of explanations, rather than primarily advanced theory Allocated time totals approximately 106 minutes; reserve additional checking time

English passage 9 (37% down the source page) 32 20 12 16 4 questions /80 marks /120 minutes Approximately per question 3-8 substantive steps; numerical questions are solvable by hand / Look up a table Each paper contains definitions / prove / modelling / a mixture of explanations, rather than primarily advanced theory Allocated time totals approximately 111 minutes; reserve additional checking time

English passage 10 (41% down the source page) 32 20 12 16 4 questions /80 marks /120 minutes Approximately per question 3-8 substantive steps; numerical questions are solvable by hand / Look up a table Each paper contains definitions / prove / modelling / a mixture of explanations, rather than primarily advanced theory Allocated time totals approximately 110 minutes; reserve additional checking time

English passage 11 (46% down the source page) number 1 Question-by-question equivalent-difficulty basis for the papers

English passage 13 (52% down the source page) 1(a) 3 A3 2025 Q1(a)(i); 2026 Q1(a) finite probability-space construction At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 14 (54% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 15 (56% down the source page) 1(b)(ii) 4 B1/C1/D2 2025 Q2(a)(ii) position-pattern probability At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 16 (58% down the source page) 1(c) 5 A2/C3 2026 Q1(c); 2020 Q2(c)(i) support-aware finite/infinite geometric sums At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 17 (60% down the source page) 1(d) 3 A2/B1 2026 Q2(c)(i) indicator-to-binomial modelling At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 3 minutes

English passage 18 (61% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 19 (64% down the source page) 2(b) 4 A2/B2 2025 Q2(b)(ii); 2023 Q3(b)(ii) marginalization on triangular support At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 20 (66% down the source page) 2(c) 3 A1/B2 2025 Q2(b)(iii); 2022 Q3(c)(iii) conditional-density ratio and support At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 21 (68% down the source page) 2(d) 2 A2 2025 Q2(b)(iv); 2023 Q3(b)(iii) support/factorization test for independence At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 3 minutes

English passage 22 (69% down the source page) 2(e) 5 B1/C1/D3 2024 Q2(b)(vi); 2022 Q2(b) joint LOTUS moments and covariance At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 23 (71% down the source page) 2(f) 3 D3 2021 Q3(c)(iii); 2023 Q3(b)(iv) CDF method for a ratio on triangular support At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 24 (73% down the source page) 3(a) 3 A3 2024 Q4(a); 2026 Q3(a) mean/median loss minimizers At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 3 minutes

English passage 25 (75% down the source page) 3(b) 3 A2/B1 2023 Q4(a); 2026 Q3(b) bias-variance identity At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 26 (76% down the source page) 3(c) 4 A2/B2 2020 Q6(a)(ii); 2024 Q4(c)(i) unknown-variance t confidence interval At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 27 (78% down the source page) 3(d) 5 A2/B1/C2 2021 Q5(b)(i)-(iii); 2026 Q3(d)(ii) one-sample t test and p-value At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 28 (80% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 29 (83% down the source page) 4(a) 4 A2/B2 2023 Q5(c); 2026 Q4(a) covariance algebra for linear combinations At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 30 (84% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 31 (87% down the source page) 4(c) 4 A1/C1/D2 2021 Q6(c); 2025 Q4(d); 2026 Q4(b) residual diagnosis and transformation At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 32 (89% down the source page) 4(d) 5 C1/D4 2026 Q4(f) support-dependent non-unique MLE At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

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 2 (10% down the source page) bootstrap sample definition At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 3 minutes

English passage 3 (14% down the source page) number 2 Question-by-question equivalent-difficulty basis for the papers

English passage 5 (20% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 6 (23% down the source page) 1(b) 3 A2/B1 2025 Q1(b)(ii) Bayes reversal after total probability At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 7 (24% down the source page) 1(c) 4 B1/C1/D2 2025 Q1(b)(iii) Bayes under a changed mechanism At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 8 (26% down the source page) 1(d) 5 A2/B2/C1 2025 Q2(a)(i) multitype password count At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 9 (28% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 10 (31% down the source page) 2(a) 4 A3/B1 2025 Q3(a)(i) inverse-transform/CDF method At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 11 (32% down the source page) 2(b) 4 A3/B1 2025 Q3(a)(ii) order-statistic CDF At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 12 (34% down the source page) 2(c) 4 A1/B1/C1/D1 2025 Q3(b)(i) MGF derivation by completing the square At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 13 (36% down the source page) 2(d) 4 A1/B1/C1/D1 2025 Q3(b)(ii) MGF product and distribution identification At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 14 (37% down the source page) 2(e) 4 B1/C1/D2 2025 Q3(c)(i) moment determinacy via MGF existence At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 15 (39% down the source page) 3(a) 2 A2 2025 Q4(a)(i) variance of an independent sample mean At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 3 minutes

English passage 16 (41% down the source page) 3(b) 6 A1/B2/C1/D2 2025 Q4(a)(ii) expectation of sample variance under unequal means At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 9 minutes

English passage 17 (43% down the source page) 3(c) 4 A3/B1 2025 Q4(b)(i) Student-t confidence interval At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 18 (44% down the source page) 3(d) 4 A2/B1/C1 2025 Q4(b)(ii) chi-square variance interval At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 19 (46% down the source page) 3(e) 4 A1/C1/D2 2025 Q4(d); 2023 Q6(d)(ii) Q-Q diagnostic At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 20 (48% down the source page) 4(a) 5 A1/B1/C1/D2 2025 Q6(a) two-boundary support-dependent MLE At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 21 (50% down the source page) 4(b) 4 A1/B1/C1/D1 2025 Q6(b)(i)-(iii) regression-estimator covariance At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 22 (51% down the source page) 4(c) 4 A1/B2/C1 2025 Q6(c) percentile bootstrap interval At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 23 (53% down the source page) 4(d) 7 A2/B1/C1/D3 2025 Q5(b) known-variance two-sample z test with p-value At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 10 minutes

English passage 24 (56% down the source page) number 3 Question-by-question equivalent-difficulty basis for the papers

English passage 26 (62% down the source page) 1(a) 5 A3/B2 2023 Q1(a) birthday collision complement At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 27 (64% down the source page) 1(b) 5 A1/B1/C1/D2 2023 Q1(b) exact single-collision count At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 28 (65% down the source page) 1(c) 5 A3/B2 2023 Q1(c) diagnostic Bayes calculation At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 29 (67% down the source page) 1(d) 5 A1/B1/C1/D2 2023 Q1(d) measurability of a piecewise discrete variable At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 30 (69% down the source page) 2(a) 4 A2/C1/D1 2023 Q3(a) dependence counterexample for Poisson sums At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 31 (70% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 32 (73% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 33 (76% down the source page) Poisson splitting via joint pmf At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 34 (79% down the source page) 3(a) 8 A5/B2/D1 2023 Q3(b)(i)-(iii) curved-support normalization and marginalization At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 11 minutes

English passage 35 (81% down the source page) 3(b) 5 A1/B1/C1/D2 2023 Q3(b)(iv) conditional CDF on curved support At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 36 (82% down the source page) 3(c) 7 A3/B2/C1/D1 2023 Q3(c) monotone CDF transform plus LOTUS At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 9 minutes

English passage 37 (84% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 10 minutes

English passage 38 (87% down the source page) 4(b) 5 A2/B1/C1/D1 2024 Q5(b) exact hypergeometric tail test At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 39 (89% down the source page) 4(c) 4 A1/B1/C2 2020 Q6(a)(i); 2023 Q4(c)(i) known-variance z confidence interval At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 40 (90% down the source page) 4(d) 3 A1/B1/C1 2023 Q4(c)(ii); 2024 Q6(a) frequentist confidence interpretation At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

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) number 4 Question-by-question equivalent-difficulty basis for the papers

English passage 3 (14% down the source page) 1(a) 3 A3 2020 Q1(a) sigma-algebra definition At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 3 minutes

English passage 4 (16% down the source page) 1(b) 3 A3 2020 Q1(b) probability-measure definition At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 3 minutes

English passage 5 (17% down the source page) 1(c) 4 A2/B1/D1 2020 Q1(e) positive stars and bars At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 6 (19% down the source page) 1(d) 4 A1/B2/C1 2022 Q1(b)(i) binomial identity proof At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 7 (21% down the source page) 1(e) 6 B2/C1/D3 2022 Q1(b)(ii) double counting/story proof At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 9 minutes

English passage 8 (23% down the source page) 2(a) 4 A4 2020 Q2(b) piecewise discrete CDF At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 9 (24% down the source page) 2(b) 6 A2/B2/C1/D1 2020 Q2(c)(ii) minimum of independent geometric waits At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 10 (26% down the source page) 2(c) 6 A1/B1/C1/D3 2020 Q2(d) pattern waiting by states/geometric stages At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 11 (28% down the source page) 2(d) 4 A1/B2/C1 2020 Q3(d)(ii) proof of total expectation At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 12 (29% down the source page) 3(a) 6 A3/B2/C1 2022 Q4(a)(i)-(ii); 2020 Q4(a)(iv)-(v) bias/variance/MSE of an estimator At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 13 (31% down the source page) 3(b) 4 A2/B2 2020 Q4(b) indicator proof of Markov inequality At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 14 (33% down the source page) 3(c) 5 A1/B1/C1/D2 2022 Q4(b); 2023 Q4(b) distribution-free sample-size guarantee At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 15 (35% down the source page) 3(d) 5 A2/B1/C1/D1 2020 Q4(c); 2022 Q5(c) Bonferroni correction At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 16 (36% down the source page) 4(a) 6 A3/B2/C1 2020 Q5(b); 2023 Q6(b) OLS derivation At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 9 minutes

English passage 17 (38% down the source page) 4(b) 5 A1/B1/C1/D2 2020 Q5(d) boundary MLE At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 18 (40% down the source page) 4(c) 4 A1/B1/C1/D1 2020 Q6(c); 2022 Q6(e) related conjugacy method Gamma conjugate update At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 19 (42% down the source page) 4(d) 5 A2/C1/D2 2020 Q6(d) non-trivial variance lower bound At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 20 (44% down the source page) number 5 Question-by-question equivalent-difficulty basis for the papers

English passage 22 (50% down the source page) 1(a) 2 A2 2023 Q2(a)(i) indicator/preimage identity At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 3 minutes

English passage 23 (52% down the source page) 1(b) 6 A2/B2/D2 2023 Q2(a)(ii) preimage sigma-algebra proof At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 24 (54% down the source page) 1(c) 4 A2/B1/C1 2026 Q2(a) pairwise-versus-mutual independence construction At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 25 (56% down the source page) 1(d) 4 A2/B1/C1 2020 Q2(a); 2022 Q1(a)(ii) finite-space measurability check At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 26 (57% down the source page) 1(e) 4 A2/B2 2021 Q1(b)(i)-(iii) product probability space and power set At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 27 (59% down the source page) 2(a) 7 A1/B1/C2/D3 2021 Q2(d) MGF of a scale mixture At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 10 minutes

English passage 28 (61% down the source page) 2(b) 6 A2/B2/D2 2022 Q2(c)(i) continuous-discrete mixture transform At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 9 minutes

English passage 29 (63% down the source page) 2(c) 7 A2/B1/C1/D3 2020 Q3(c) three-dimensional normalization and LOTUS At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 10 minutes

English passage 30 (64% down the source page) 3(a) 5 A2/B1/C1/D1 2024 Q5(b) variant exact hypergeometric test At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 31 (66% down the source page) 3(b) 4 A2/B1/C1 2023 Q4(d) quartiles and Tukey outliers At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 32 (68% down the source page) 3(c) 3 A1/B1/C1 2021 Q6(d); 2023 Q5(d) Simpson reversal/confounding At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 33 (69% down the source page) 3(d) 5 A1/B1/C1/D2 2021 Q4(b) Chebyshev sample-size design At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 34 (71% down the source page) 3(e) 3 A1/B1/C1 2022 Q5(a); 2025 Q5(c) testing error trade-off At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 35 (73% down the source page) 4(a) 6 A2/B1/C1/D2 2022 Q6(a) regular log-likelihood MLE At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 36 (75% down the source page) 4(b) 4 A3/B1 2022 Q6(d) OLS mean-point property At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 37 (76% down the source page) 4(c) 4 A1/B1/C1/D1 2025 Q3(c)(i) moment determinacy At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 38 (78% down the source page) 4(d) 3 A2/B1 2025 Q6(c) bootstrap percentile interval At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 39 (80% down the source page) 4(e) 3 A2/B1 2021 Q4(a)(iii) constructing an unbiased estimator At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 40 (83% down the source page) number 6 Question-by-question equivalent-difficulty basis for the papers

English passage 42 (89% down the source page) 1(a) 6 A2/B2/C1/D1 2022 Q1(b) method, altered parameter algebraic and narrative double counting At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 9 minutes

English passage 43 (90% down the source page) 1(b) 6 A2/B2/C1/D1 2022 Q3(a); 2024 Q1(c) Bayes/base-rate transfer At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

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 2 (10% down the source page) 1(c) 5 A1/B1/C1/D2 2023 Q1(b) exact birthday collision At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 3 (12% down the source page) 1(d) 3 A3 2025 Q1(a)(i); 2021 Q1(b) finite product probability space At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 3 minutes

English passage 4 (13% down the source page) 2(a) 8 A4/B2/C1/D1 2022 Q3(c)(i)-(iii) conditional expectation on triangular support At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 11 minutes

English passage 5 (15% down the source page) 2(b) 7 A2/B2/C1/D2 2024 Q3(b)(ii)-(iii) compound-Poisson moments At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 9 minutes

English passage 6 (17% down the source page) 2(c) 5 A2/B1/C1/D1 2025 Q3(a); historical transformation chain order statistic followed by monotone transform At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 7 (18% down the source page) 3(a) 6 A1/B2/C1/D2 2025 Q4(a)(i)-(ii) non-identical-normal sample variance expectation At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 8 minutes

English passage 8 (20% down the source page) 3(b) 7 A2/B2/C1/D2 2023 Q5(b) pooled two-sample t test for a linear combination At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 10 minutes

English passage 9 (22% down the source page) 3(c) 4 A2/B1/C1 2026 Q3(d)(i) chi-square variance interval At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 10 (24% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 5 minutes

English passage 11 (26% down the source page) 4(a) 5 A2/B1/C1/D1 2024 Q6(c) least squares for a transformed predictor At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 12 (28% down the source page) At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

English passage 13 (31% down the source page) 4(c) 5 A2/B1/C1/D1 2026 Q4(f) non-unique support MLE At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 7 minutes

English passage 14 (33% down the source page) 4(d) 4 A1/B1/C1/D1 2021 Q6(b) Beta-geometric conjugacy At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 6 minutes

English passage 15 (34% down the source page) 4(e) 3 A2/B1 2024 Q5(d); 2026 Q4(d) spurious correlation interpretation At equal marks, calibrate steps, computation and abstraction against similar official questions; topic similarity is not difficulty evidence . 4 minutes

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