Topic Brief: Discrete Random Variables Probability Function

#SIGNAL Topic Master Brief — Discrete Random Variables: The Probability Function (WST01)

Provenance: written 2026-09-29, built entirely on research/veridian/WST01-verified-facts.md (§1, §3, §5, §7, §8, and the per-lesson audit entries for discrete-random-variables-probability- function.ts and expectation-and-variance-of-linear-transformations.ts), cross-checked against the existing signal-frequency-and-boundaries.md, codex-deep-dive-discrete-random-variables-probability- function.ts and forge-exam-technique-guide.md in this same folder, and against content/veridian/curriculum/wst01/discrete-random-variables-probability-function.ts. No new PDF fetches, web searches, or primary-source downloads were performed for this document. Every claim below traces to one of those already-verified sources, cited by series and question number.

#Scope: narrower than "discrete random variables"

signal-frequency-and-boundaries.md reports one combined row for spec 5.1–5.4 ("5 of 6" ER-read series). That row includes spec 5.3 — the E(aX+b)/Var(aX+b) mean-variance shortcuts — which is a separate curriculum topic (expectation-and-variance-of-linear-transformations.ts) with its own frequency history. This brief narrows to exactly the scope of the lesson it's paired with: 5.1 (the concept), 5.2 (the probability function and CDF), and 5.4 (the discrete uniform distribution) — deliberately excluding 5.3. That narrowing changes the honest count, which is the whole reason this document exists rather than just re-citing the existing "5 of 6" figure.

#1. Every real series confirmed testing this specific sub-scope

Borderline, not counted above: June 2019 Q5(b)–(c) uses the sum-to-1 property as one line of working to solve for an unknown probability $q$, but the question's dominant, mark-heavy content is spec 5.3 (E(X), Var(X), and — its real point — E(X) versus P(win)), and it is filed under that spec tag in the lesson that teaches it. Listed here for completeness, not folded into the count below.

Checked and confirmed as a different sub-scope, not this one: Jan 2021 Q4(b)–(d) and Jun 2024 Q2(b)–(c)/Q3(d) are all confirmed discrete-random-variable content, but specifically the "forgot-to-square" Var(aX+b) error (spec 5.3) — not this topic. Jan 2023 Q3/Q4 (11 and 13 marks) are confirmed as discrete-random-variable questions in general, but the facts bank and the FORGE guide both flag that their sub-item content was never itemised (forge-exam-technique-guide.md §1) — they cannot honestly be added to either topic's count.

The honest frequency statement: this exact sub-scope (5.1/5.2/5.4) is seen in 2 of the 6 series with a fully-read examiner report (Oct 2021, Jun 2022), by direct examiner-report quote — plus one further ER-read series (Jun 2024) where a 1-mark instance is confirmed by the mark scheme alone, not discussed in that series' examiner report — plus one series outside that 6-series denominator entirely (Jan 2025, no examiner report located) confirmed via a full mark-scheme worked example. Examiner reports for 8 of the 14 mark-scheme series were not located (Jun 2021, Jan 2022, Oct 2022, Jun 2023, Oct 2023, Jan 2024, Oct 2024, Jan 2025) — this is a lower bound, not an exhaustive count.

#2. Real mark-tariff pattern

Two shapes, not one. In three of the four confirmed instances, this sub-scope is a low-mark sub-part (1–2 marks) embedded inside a larger discrete-random-variable question whose other parts test something else: Oct 2021 Q4(d), Jun 2022 Q5(d), and Jun 2024 Q2(a) (1 mark, B1) are each single parts of multi-part questions, not questions of their own. The discrete uniform distribution specifically has never anchored a multi-part question by itself in the reviewed record — naming it is consistently a single opening mark (Jun 2024 Q2(a); Jan 2025 Q1(a)).

The exception is Jan 2025 Q1, a 7-question series where this single question carries 15 of the paper's 75 marks — one-fifth of the entire exam — spanning nearly the whole 5.1/5.2/5.4 sub-scope (plus 3.1) in one go. No comparable full-question concentration on this narrow sub-scope is confirmed in any other series. Position on the paper is inconsistent, not a pattern: Jan 2025's instance opens the paper (Q1); the borderline June 2019 case sits second-to-last (Q5 of 6, 14 marks total for the whole question). No safe claim about where in the paper this topic lands can be made from four data points spread across "opening question" and "second-to-last question."

#3. Recurring context/angle type

• Chance-device framing. Every confirmed real instance uses a spinner or dice, never a real dataset: Jan 2025's two four-sided dice, Jun 2024's spinner, the borderline June 2019 biased spinner. This sub-scope is tested through constructed probability models, not survey/measurement data — unlike, for contrast, Section 2's data-representation topic.
• Two angle types on the $\sum p(x)=1$ property specifically, confirmed from opposite sides: checking a function that looks complete but isn't (Oct 2021 — a value left out of the domain) and constructing one from an unknown constant (Jun 2022 — the equation itself is the mark-earning step, separate from solving it).
• CDF-inversion (recovering a missing table entry from the entries around it) has exactly one confirmed real instance (Jan 2025 Q1(g)) — a genuine technique, not yet a confirmed recurring pattern on this paper.

#If this topic appears, here's what to expect

Most likely: a short, low-mark sub-part (1–2 marks) inside a bigger discrete-random-variable question — either naming or using a probability function correctly, with the domain-completeness check ($\sum p(x)=1$) the single most-evidenced trap on this exact sub-scope. Less likely but confirmed at least once: this sub-scope carrying an entire opening question worth up to a fifth of the paper's marks, built around a dice or spinner scenario and touching the concept, the probability function, the discrete uniform distribution, and CDF-inversion together. Not confirmed anywhere in the record: a multi-part question built around the discrete uniform distribution alone, or a second CDF-inversion instance.

#Sources and limitations

Built entirely from research/veridian/WST01-verified-facts.md §1 (spec text), §3 (source archive — 6 of 14 series have a located examiner report), §5 (Section 5 per-topic content), §7–§8 (topic strength, flagged gaps), and the two per-lesson audit entries for this lesson and its 5.3 sibling. No grade-boundary or likelihood figure is stated anywhere above, consistent with the facts bank's own §2 confirmation that no such data exists in the sources used for this course. Examiner reports for 8 of 14 series remain unread; the counts above are a lower bound on this sub-scope's real testing history.