Technique Masterclasses

Statistics 1

Topic Brief: Expectation and Variance of Linear Transformations

SIGNAL Topic Master Brief — Expectation and Variance of Linear Transformations (WST01, spec 5.3)

Provenance (2026-09-29): built entirely from research/veridian/WST01-verified-facts.md (compiled 2026-09-12, spot-check addendum 2026-09-25), grepped for "aX+b", "linear transformation" and "5.3". Cross-checked against signal-frequency-and-boundaries.md (this folder, whole-Section-5 "5 of 6" figure, which pools spec 5.1–5.4 together) and codex-deep-dive-expectation-and-variance-of-linear-transformations.md (this folder, formal derivations and sub-case structure) so this document does not restate either — it narrows the frequency doc's pooled discrete-RV count down to this one shortcut-formula pairing and adds the mark-tariff/context-pattern layer neither existing document treats as its own subject. No new PDF fetches were made. Every claim below cites a series and question from the facts bank.

1. Frequency — seen in 2 of 6 series checked

The facts bank has a fully-read Principal Examiner Feedback report for 6 of the 14 WST01 series with a verified mark scheme: Jun 2019, Jan 2021, Oct 2021, Jun 2022, Jan 2023, Jun 2024. Within that set, the specific error this topic is known for — misapplying E(aX+b)=aE(X)+b or Var(aX+b)=a²Var(X) — is directly evidenced in 2 of those 6: Jan 2021 and Jun 2024, with Jun 2024 contributing two separate questions in the same sitting (Q2(c) and Q3(d)).

Worth flagging plainly: the facts bank's own §5 prose describes this as "confirmed independently in 3 separate series," but the three citations it then lists resolve to only two distinct series (Jan 2021 once, Jun 2024 twice). This brief follows the citation list, not the summary sentence, and the course's own CODEX deep-dive on this exact topic independently lands on the same "2 of 6" figure. Two real series, three real question-instances — stated as such, not rounded up.

The other 8 series were not searched question-by-question for this specific shortcut-formula error; their absence from the count is not evidence they lacked it. This is narrower than the pooled "5.1–5.4 Discrete random variables, 5 of 6" row in signal-frequency-and-boundaries.md — that row credits any discrete-RV content (probability functions, uniform distribution, this topic) across five series; this brief isolates only the aX+b shortcut itself.

2. What was actually tested, by series

3. Mark-tariff pattern

The two real anchors that give a full worked structure sit at different tariffs: Jun 2024 Q2(c) is worth 4 marks inside a 12-mark question (only Q2(b)–(c), 7 marks, sit inside spec 5.3; Q2(a) is spec 5.2, Q2(d) is spec 3); Jan 2021 Q4(d) is worth 5 marks alone. Both force the same equation order: the variance equation (Var(Y)=a²Var(X)) must be solved first, since only it isolates the multiplicative constant a; only then can the mean equation (E(Y)=aE(X)+b) give the additive constant b. Neither real question solves mean-equation-first. Individual mark values for Jan 2021 Q4(b)–(c) and Jun 2024 Q3(d) are not stated anywhere in the record, so none is claimed. Both backward-solve anchors sit as a later question on their paper (Q4; Q2/Q3) — with only two series, that is not enough to generalise a "always late" rule, only to note no early-paper example has turned up yet.

4. Recurring context/angle across the two series

Both real anchors follow one identical structure, not a family of setups: a base discrete random variable X, a second variable defined as a linear transformation of it (Y=aX−b or W=aX+b), the transformed variable's own mean and variance given as targets, and the task is to find a and b — variance first, mean second. The headline error recurs three times across those two series in the same shape: forgetting the square in Var(aX+b)=a²Var(X). Jan 2021 additionally surfaces two less-mechanical failures on the same question — E(X²) mistaken for Var(X), and the shortcut formulae simply not known — that a "don't forget to square" lesson alone would not catch.

One adjacent, out-of-scope pattern is worth naming, not folding in: Jun 2022 Q3(e) (a coding question, spec 2.2) shows the identical missing-square mechanism decoding a variance the other direction — "scaling using 0.5 instead of 0.5²". It is not counted in the "2 of 6" above, being a different spec point, but it confirms the error is a property of the arithmetic, not of how spec-5.3 questions happen to be phrased.

5. If this topic appears, here's what to expect

Expect a two-part structure: a "show that Var(X)=[value]" step on the base variable, followed by a second variable defined as a linear transformation of the first, with its mean and variance given as targets and a, b (or equivalent constants) to find. Solve the variance equation first — it is the only one that isolates the multiplicative constant — and only then substitute into the mean equation for the additive one; both real anchors are unsolvable in the other order. The single most common recorded error, three times over two series, is dropping the square on a in the variance equation; a close second, seen once, is confusing E(X²) with Var(X) upstream of that step. If the constants come out as a decimal or a negative number (as in the Jan 2021 anchor, a=0.6, b=−4.8), that is not itself a sign of an error — both real mark schemes accept non-integer, negative solutions.

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