Statistics 1

Deep Dive: Expectation and Variance of Linear Transformations

#CODEX Deep-Dive — Expectation and Variance of Linear Transformations (WST01 §5.3)

*Provenance, 2026-09-29: written after reading research/veridian/WST01-verified-facts.md in full (§1 spec table, §2a formula-booklet contents, §2b AO weighting, §5 Section-5 discrete-RV findings, §6 cross-cutting "show that" material, and the file's own 2026-09-12 audit-log entry for this exact lesson) and cross-checking it against the live lesson file content/veridian/curriculum/wst01/expectation-and-variance-of-linear-transformations.ts (read in full), the three existing technique-hub documents in content/veridian/technique-hub/wst01/ (codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md), and content/veridian/question-bank/wst01.ts (grepped — its one spec-5.3 item concerns a different trap, E(X)>0 versus a favourable win-probability, and is correctly out of scope here; noted in §6). No new PDFs, web fetches, or external searches were performed. Everything below traces to one of those documents by name.

#Why this document exists alongside the lesson file

The taught lesson already derives both shortcut formulae from first principles, works a real backward-solving question (Jun 2024 Q2(c)) end to end for both unknowns, drills a parallel chain-drill, and carries a marked-solution, three trap-taxonomy items and five MCQs. This document does not re-teach any of that. It exists to do three things the lesson's pedagogical shape has no room for: lay the complete sub-case structure of the linear-transformation rule out as one reference (rather than as a single narrative derivation), connect it explicitly to the two other spec points where the same mechanism resurfaces under a different name (coding, §2.2; PMCC invariance, §4.3), and give the exam-angle synthesis its own honest accounting — including a second real backward-solving anchor, Jan 2021 Q4(d), whose own numbers the lesson's audit trail explicitly logged as read and deliberately left unused.

#1. The complete real model

#1.1 The one formula everything is built from

The Mathematical Formulae and Statistical Tables booklet ("Statistics S1" section, verified in WST01-verified-facts.md §2a) prints exactly three general-purpose definitions for a discrete random variable $X$ : $E(X)=\mu=\sum x_iP(X=x_i)$ ; $\text{Var}(X)=\sigma^2=\sum(x_i-\mu)^2P(X=x_i)=\sum x_i^2P(X=x_i)-\mu^2$ ; and the general expectation-of-a-function rule $E(g(X))=\sum g(x_i)P(X=x_i)$ . Every result in this document is that third formula with a specific $g$ substituted in — nothing in what follows is a separate axiom.

#1.2 Deriving $E(aX+b)=aE(X)+b$

Set $g(x)=ax+b$ . Then $E(aX+b)=\sum(ax_i+b)P(X=x_i)$ . Because $a$ and $b$ are constants, not random quantities, the sum splits: $a\sum x_iP(X=x_i)+b\sum P(X=x_i)$ . The first term is $aE(X)$ by definition; the second is $b$ times the total probability, which is $1$ for any discrete distribution (spec 5.2), so it collapses to $b$ . Result: $E(aX+b)=aE(X)+b$ — linearity, and nothing more exotic.

#1.3 Deriving $\text{Var}(aX+b)=a^2\text{Var}(X)$

Variance is not itself a value of $g(X)$ summed directly — it is $E\!\left[(X-\mu)^2\right]$ , built from the squared deviation from the mean, so the mean of $aX+b$ has to be found first. From §1.2 that mean is $aE(X)+b$ . The deviation of $aX+b$ from its own mean is therefore $(aX+b)-(aE(X)+b)=a\big(X-E(X)\big)$ — note $b$ has already cancelled, before any squaring happens. Squaring that deviation gives $a^2\big(X-E(X)\big)^2$ , and taking the expectation pulls the constant $a^2$ back out: $\text{Var}(aX+b)=a^2E\!\left[(X-E(X))^2\right]=a^2\text{Var}(X)$ .

#1.4 The sub-case table this collapses into

The general rule has three genuinely distinct special cases, each of which is a legitimate exam question shape in its own right, not merely an algebraic curiosity:

Sub-case	Transformation	$E$ result	$\text{Var}$ result	Micro-example
Pure shift	$Y=X+b$ ( $a=1$ )	$E(X)+b$	$\text{Var}(X)$ — unchanged	$E(X)=5$ , $\text{Var}(X)=2$ , $Y=X+10 \Rightarrow E(Y)=15$ , $\text{Var}(Y)=2$ . Every outcome slid by the same 10; the gaps between them, which is all variance measures, are untouched.
Pure scale	$Y=aX$ ( $b=0$ )	$aE(X)$	$a^2\text{Var}(X)$	$E(X)=5$ , $\text{Var}(X)=2$ , $Y=3X \Rightarrow E(Y)=15$ , $\text{Var}(Y)=18$ (not $6$ ).
Combined	$Y=aX+b$	$aE(X)+b$	$a^2\text{Var}(X)$	$E(X)=5$ , $\text{Var}(X)=2$ , $Y=3X+10 \Rightarrow E(Y)=25$ , $\text{Var}(Y)=18$ — the shift changes $E(Y)$ again but still contributes nothing to $\text{Var}(Y)$ .
Decode (inverse direction)	Given $\text{Var}(Y)$ where $Y=aX+b$ , recover $\text{Var}(X)$	$E(X)=\dfrac{E(Y)-b}{a}$	$\text{Var}(X)=\dfrac{\text{Var}(Y)}{a^2}$ — divide by $a^2$ , never multiply	$\text{Var}(Y)=18$ , $a=3 \Rightarrow \text{Var}(X)=18\div9=2$ .

Every numeric value in this table is a fresh, hand-checked VERIDIAN-original illustration built to show the boundary between sub-cases side by side — none is a past-paper reproduction. The pure-shift and pure-scale rows are the two halves the lesson's derivation already proves together; this table's only job is to make each one individually checkable in isolation, plus name the decode direction as its own row, since that direction is where a distinct real error (§2 below) actually lives.

#1.5 What this does not extend to

$E(X^2)\neq\big[E(X)\big]^2$ in general — squaring is not itself a linear transformation ( $g(x)=x^2$ is not of the form $ax+b$ ), so none of the derivation above applies to it. This is not a side note: the verified record shows candidates on this exact spec point conflating the two directly. Examiner commentary on Jan 2021 Q4 records that some candidates believed $E(X^2)=\text{Var}(X)$ — collapsing two different quantities (a raw second moment, and that moment with $[E(X)]^2$ already subtracted back off) into one. This sits one step upstream of the linear-transformation rule itself: get $\text{Var}(X)$ wrong via this conflation and every $a^2\text{Var}(X)$ calculation built on top of it inherits the error.

#1.6 A direct mathematical consequence, not independently exam-verified

Square-rooting §1.3 gives $\text{SD}(aX+b)=|a|\cdot\text{SD}(X)$ — the absolute value is required because a standard deviation cannot be negative even when $a$ is. This is a true, checkable consequence of the variance rule and is stated here only as that: nothing in the verified-facts record shows a WST01 question testing standard deviation under a transformation directly (as opposed to testing variance and leaving the candidate to root it). It is included for completeness of the real model, flagged explicitly as derived rather than exam-attested, per this document's own sourcing discipline.

#1.7 The memorise/booklet boundary, stated once

The spec's own "Notation and formulae" box states plainly that formulae students are expected to know "will not appear in the booklet" (verified, WST01-verified-facts.md §2a) — and $E(aX+b)=aE(X)+b$ , $\text{Var}(aX+b)=a^2\text{Var}(X)$ are named explicitly on that expected-to-know list, while the three base definitions in §1.1 above are printed in full. The practical consequence is asymmetric risk: a candidate who forgets $E(X)=\sum x_iP(X=x_i)$ can recover it from the booklet in the exam room; a candidate who forgets the $a$ , $a^2$ shortcut pattern has nowhere to look it up and must reconstruct it from §1.2–1.3 live, under time pressure, or lose the mark outright.

#2. The same mechanism under a different name: coding (spec 2.2)

Spec item 2.2 ("understanding and use of coding") is a different content area — summarising raw data, not discrete probability distributions — but a linear coding step $y=\dfrac{x-a}{b}$ or $y=b(x-a)$ is algebraically the same $aX+b$ transformation run in reverse, and the verified record shows the identical failure mode surfacing there independently. Examiner commentary on Jun 2022 Q3(e) (log-length data, coded variable $W$ ) records that correct variance answers were rare, with candidates not decoding at all, or decoding the mean but "scaling using 0.5 instead of 0.5²" — the same missing-square error as §1.3, arrived at from the data-summary side of the spec rather than the discrete-RV side. This confirms the trap is not an artefact of how discrete-RV questions happen to be phrased; it is a property of the underlying arithmetic, and it recurs wherever a linear rescaling and its inverse are both required in one question, regardless of which spec section supplies the cover story.

#3. What does not change under a linear transformation: PMCC invariance (spec 4.3 contrast)

A genuinely different real exam fact worth holding up against §1's transformation rules, because it draws the boundary of what "linear transformation" affects and what it leaves alone. Examiner-report material on Jan 2024 Q2(d) confirms Pearson's own credited point that $r$ is "not affected by (linear) coding" — the product moment correlation coefficient is invariant under a linear rescaling of either variable. This is not a coincidence needing its own memorisation; it falls out of the booklet's own $S_{xx}$ , $S_{yy}$ , $S_{xy}$ formulae (verified, WST01-verified-facts.md §2a) by the identical substitution method as §1.2–1.3. If $X'=aX+b$ with $a>0$ , every deviation from the mean scales by $a$ , so $S_{x'x'}=a^2S_{xx}$ and $S_{x'y}=aS_{xy}$ (only one of the two deviations in the cross-product scales). Substituting into $r=S_{xy}/\sqrt{S_{xx}S_{yy}}$ :

r'=\frac{aS_{xy}}{\sqrt{a^2S_{xx}\cdot S_{yy}}}=\frac{aS_{xy}}{|a|\sqrt{S_{xx}S_{yy}}}=\text{sign}(a)\cdot r.

For $a>0$ (the case actual coding always uses) $r'=r$ exactly — the credited fact. The algebra also shows, as a direct mathematical extension beyond what the cited exam material itself states, that a negative scaling constant would flip $r$ 's sign rather than leave it invariant; no verified WST01 question tests that negative- $a$ case, so it is flagged here as derived, not attested. The contrast with §1 is the real teaching point: $E$ and $\text{Var}$ of a single variable transform under $aX+b$ exactly as §1 describes, while a relationship between two variables can be completely untouched by the same transformation — two different objects, two different transformation behaviours, easy to blur if the word "coding" is treated as one undifferentiated topic rather than as an operation whose effect depends on what quantity it's being applied to.

#4. The tested question shape: solving backwards for two unknowns

Both real anchors the verified-facts record attaches to this spec point ask a candidate to find two unknown constants, not one — and the two constants each come from a different one of §1's two equations, solved in a forced order. The lesson's own worked-chain already reproduces the first anchor (Jun 2024 Q2(c): $E(X)=4.6$ , $\text{Var}(X)=4.14$ , $Y=aX-b$ , $E(Y)=13.4$ , $\text{Var}(Y)=66.24 \Rightarrow a=4$ , $b=5$ ) in full, so it is not repeated here.

A second real anchor exists in the verified-facts record and was explicitly read but never written into the taught lesson (its own audit-log entry names this exactly: the numbers were "not written into the lesson," used only as corroboration). It is reproduced here for the first time as a short, standalone micro-example, since it is a genuinely different-shaped instance of the same question type — a non-integer multiplicative constant and a negative additive constant, where the first anchor's were both positive integers:

Jan 2021 Q4(d) (5 marks): $W=aX+b$ , $a>0$ , given $E(X)=13$ , $\text{Var}(X)=5$ , $E(Y)=3$ , $\text{Var}(Y)=1.8$ , and $E(W)=E(Y)$ , $\text{Var}(W)=\text{Var}(Y)$ , find $a$ and $b$ .

Variance equation first, since it alone determines $a$ : $a^2(5)=1.8 \Rightarrow a^2=0.36 \Rightarrow a=0.6$ (taking the positive root, as given). Mean equation second, now that $a$ is known: $3=0.6(13)+b \Rightarrow 3=7.8+b \Rightarrow b=-4.8$ .

Both figures ( $a=0.6$ , $b=-4.8$ ) match the verified-facts record exactly and were re-derived independently for this document rather than copied through unchecked. Two real anchors, three years apart, both requiring the identical two-equation, forced-order structure ("solve the variance equation for the multiplicative constant; only then can the mean equation be solved for the additive one") with different-shaped numbers is stronger evidence that this is the genuine tested question shape for this spec point, not an artefact of one series' paper design.

#5. Synthesis: every real exam angle this spec point has actually been tested from

Using this course's own stated methodology (signal-frequency-and-boundaries.md): "N of 6" below means confirmed by a specific question/mark-scheme citation in N of the 6 WST01 series with a fully-read examiner report (Jun 2019, Jan 2021, Oct 2021, Jun 2022, Jan 2023, Jun 2024). Absence from a series' count is not evidence the angle was absent that series — only that no citation for it was located.

Exam angle	Confirmed in	Mechanism
Forgetting to square $a$ in $\text{Var}(aX+b)$	2 of 6 (Jan 2021; Jun 2024 — twice independently within that one series, Q2(c) and Q3(d))	The headline trap on this spec point; identical mechanism each time.
$E(X^2)$ conflated with $\text{Var}(X)$	1 of 6 (Jan 2021 Q4)	A different, upstream error feeding a wrong $\text{Var}(X)$ into any later $a^2\text{Var}(X)$ step.
Shortcut formulae not known from memory at all	1 of 6 (Jan 2021 Q4)	Distinct from the square-forgetting trap — a candidate with no formula, correct or not, to substitute into.
Decode-direction error under coding (dividing vs. multiplying by $a^2$ ; wrong assumed mean)	1 of 6 (Jun 2022 Q3(e))	Same missing-square mechanism, arrived at from spec 2.2 rather than 5.3 directly (§2 above).
Two-unknown backward solve, forced equation order	2 real anchors located (Jun 2024 Q2(c); Jan 2021 Q4(d)), spanning 3 years	Variance equation must be solved for the multiplicative constant before the mean equation can be solved for the additive one (§4 above).
PMCC invariant under linear coding	1 of 6 (via mark-scheme citation, Jan 2024 Q2(d) — outside the 6 ER series, noted separately)	Contrast case: shows what a linear transformation does not change (§3 above).

#6. Honest boundaries — what this document does not claim

Discrete uniform distribution (5.4) combined with $aX+b$ : no verified instance in the facts bank applies the linear-transformation shortcut to a named discrete uniform variable specifically (the one rich discrete-uniform anchor, Jan 2025 Q1, uses $R+B$ and a derived CDF instead — a sum of two variables and a different random variable altogether, not a linear transformation of one). Not claimed here.
Sums or differences of two random variables ( $E(X+Y)$ , $\text{Var}(X-Y)$ , independence assumptions between them): outside the WST01 spec entirely as verified in §1 of the facts bank — S1's spec item 5.3 covers a single variable's own linear transformation only. Not part of this document.
Standard deviation tested directly under a transformation (§1.6): a true derived consequence, not a claim that a real WST01 question has tested it that way — no such citation exists in the record read for this document.
A negative- $a$ sign-flip in PMCC invariance (§3): real mathematics, not a claim that a real WST01 question has tested it — flagged as a derived extension beyond the cited source.
No likelihood percentages or grade boundaries are stated anywhere above. Frequency claims use only "confirmed in N of 6 series checked" language, per this course's own standing convention (signal-frequency-and-boundaries.md §2 records that no real WST01 grade-boundary data exists in the source material at all).

#Sources

research/veridian/WST01-verified-facts.md — §1 (spec table, 5.3 guidance text), §2a (formula-booklet contents), §5 Section 4 (PMCC/coding) and Section 5 (discrete random variables, 5.3), and the 2026-09-12 audit-log entry for expectation-and-variance-of-linear-transformations.ts (Jan 2021 Q4(d) numbers).
content/veridian/curriculum/wst01/expectation-and-variance-of-linear-transformations.ts — read in full to confirm what the taught lesson already covers, so this document does not repeat it.
content/veridian/technique-hub/wst01/codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md — read/grepped for existing 5.3 coverage and for this course's own frequency-reporting convention, reused in §5 above.
content/veridian/question-bank/wst01.ts — grepped; its one spec-5.3 item is a different trap (E(X)>0 vs. a favourable win probability) and is out of scope here.