Statistics 1

Deep Dive: Discrete Random Variables Probability Function

#CODEX Deep-Dive — Discrete Random Variables: The Probability Function (WST01, spec 5.1 / 5.2 / 5.4)

Provenance (2026-09-29): written after reading research/veridian/WST01-verified-facts.md in full (§1 spec table of contents, §2a formula-booklet contents, §3 source archive, §5 Section 5, §7, §8, and the two per-lesson audit entries for discrete-random-variables-probability-function.ts dated 2026‑09‑12 and 2026‑09‑25); the full existing lesson file content/veridian/curriculum/wst01/discrete-random-variables-probability-function.ts; the three existing WST01 technique-hub documents in this folder (codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md); and content/veridian/question-bank/wst01.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 name and, where possible, by series and question number. Where the existing lesson file already reproduces a real question's full working, this document deliberately does not repeat that working — it cross-references it instead (§8) and spends its own space on material that lesson does not already cover: a second, independent derivation of why probabilities must sum to 1, the CDF-inversion skill, short constructed micro-examples for each sub-case, and a single consolidated synthesis of the real exam angles this exact topic has been tested from.

#Scope

This document covers spec items 5.1 (the concept of a discrete random variable), 5.2 (the probability function and cumulative distribution function), and 5.4 (the discrete uniform distribution). Spec item 5.3 — the mean/variance shortcut formulae E(aX+b), Var(aX+b) — is the subject of a sibling lesson, expectation-and-variance-of-linear-transformations.ts, and is only touched here where 5.4's own guidance requires it (finding the mean and variance of a specific distribution), exactly as the existing lesson file's own header already states for itself.

#1. What a discrete random variable actually is (spec 5.1)

The spec's own content list states this item in five words: "the concept of a discrete random variable." A discrete random variable $X$ is a variable whose value is the outcome of a random process and which can take only a finite, or countably infinite, list of separate values — never a value lying strictly between two of them. That separateness is what "discrete" is doing: it is the property that makes it sensible to attach one specific probability to each individual value, rather than to a whole interval of values (the latter is the continuous case, handled elsewhere on this course by the Normal distribution). Capital $X$ names the random process itself, before any outcome is known; lowercase $x$ names one particular value it might take.

The verified facts bank's own per-topic breakdown records that this spec item does not anchor a standalone question anywhere in the reviewed archive — it functions as the foundation every other Section 5 item (the probability function, the mean and variance, the discrete uniform distribution) is built directly on top of, rather than as content examined in its own right (WST01-verified-facts.md §5, Section 5 discussion).

#2. The probability function: definition and its two defining conditions (spec 5.2)

The probability function is $p(x) = P(X=x)$ : the probability that $X$ takes the specific value $x$ . The spec's own guidance keeps the examinable skill deliberately narrow — "simple uses of $p(x) = P(X=x)$ " — which the facts bank confirms means the exam mostly tests whether a probability function can be read and used correctly, not derived from first principles.

Any valid probability function must satisfy two conditions simultaneously, and both matter:

$0 \le p(x) \le 1$ for every value $x$ in the domain.
$\sum p(x) = 1$ , summed across the entire domain.

Condition 2 is where the entire documented trap history for this spec item lives (§7 below), so it is worth deriving properly rather than asserting.

#2.1 Why $\sum p(x)$ must equal exactly 1 — a second derivation

The existing lesson file already derives this from the general addition rule for mutually exclusive events (spec 3.2) collapsing to plain addition. A genuinely different, and slightly more general, route to the same fact uses indicator variables and is worth having as a second angle on the same result, because it generalises past this one spec point (it is the same argument used, later in a full A-level course, to justify why any partition of a sample space has probabilities summing to 1).

For each value $x$ in $X$ 's domain, define the indicator $\mathbb{1}_{X=x}$ , equal to $1$ if $X$ actually takes the value $x$ on a given trial and $0$ otherwise. Because $X$ takes exactly one value on any given trial — never zero of the listed values, and never two at once — exactly one of these indicators is $1$ and every other one is $0$ . That means, with certainty, on every single trial:

\sum_x \mathbb{1}_{X=x} = 1.

This is not a probabilistic statement yet — it is true every single time, by the plain fact that $X$ takes exactly one value. Taking the expectation of both sides (expectation of a sum is the sum of expectations, and $E[\mathbb{1}_{X=x}] = P(X=x) = p(x)$ for an indicator variable):

E\left[\sum_x \mathbb{1}_{X=x}\right] = \sum_x p(x), \qquad E[1] = 1.

So $\sum_x p(x) = 1$ — not approximately, not "usually," but exactly, because the left-hand quantity was already, deterministically, equal to $1$ before any probability was taken. A probability function whose values sum to anything other than $1$ is therefore not describing a genuine partition of $X$ 's possibilities: either a value has been double-counted, or — the single most commonly documented real error on this exact spec point (§7.1) — a genuinely possible value has been left out of the sum entirely, so the indicators being summed no longer cover every trial.

#3. The cumulative distribution function (spec 5.2)

The second half of spec 5.2 is the cumulative distribution function, $F(x_0) = P(X \le x_0) = \sum p(x)$ for $x \le x_0$ : a running total of probability, accumulated from the smallest value in the domain up to and including $x_0$ . Two conventions are easy to get backwards under exam pressure and are worth stating as separate, checkable rules rather than folded into the definition:

The inequality is $\le$ , not $<$ — $F(x_0)$ includes the probability that $X$ equals $x_0$ itself, not only the probability that $X$ falls strictly below it.
The sum always runs from the smallest domain value upward, never from the largest value downward.

This particular formula carries a genuine, checkable status worth flagging on its own: the verified facts bank confirms, from the spec's own "Notation and formulae" section, that $F(x_0) = P(X \le x_0) = \sum p(x)$ for $x \le x_0$ is explicitly listed among the formulae students are expected to know from memory, stated in the spec's own words as formulae that "will not appear in the booklet" (WST01-verified-facts.md §2a). Unlike the base $E(X)$ , $\mathrm{Var}(X)$ and $E(g(X))$ definitions, which are printed in the Mathematical Formulae and Statistical Tables booklet, the CDF's own definition is not — so both the $\le$ -versus- $<$ boundary and the direction of the running sum have to be recalled correctly under exam conditions, not looked up.

#3.1 Going backwards: recovering $p(x)$ from $F(x)$

A less obvious, but real, direction of this same skill runs the other way: given a table of $F$ -values (possibly with one entry missing) rather than a table of $p$ -values, an individual probability can be recovered as the difference between two consecutive cumulative totals. If the domain values are ordered $x_1 < x_2 < \dots$ , then for any $x_k$ in that ordered list,

p(x_k) = F(x_k) - F(x_{k-1}),

where $x_{k-1}$ is the next smaller value actually present in the domain (not necessarily $x_k - 1$ as a plain number, since a discrete random variable's domain need not consist of consecutive integers — see §4). This inversion is exactly the technique a real, verified WST01 question requires to fill in a missing CDF-table entry for a derived random variable, and the mark scheme for that question accepts either direction of the same idea — building the missing entry up from the value below it, or down from a value above it via the complement of a probability already found (WST01-verified-facts.md §5, Jan 2025 Q1(g), discussed further in §7.5 below; the full working is not repeated here — see §8).

#4. The discrete uniform distribution (spec 5.4)

A discrete random variable is Discrete Uniform when every value in its domain is equally likely — that is the entire definition, and nothing more. For $n$ possible values, each one carries probability $p(x) = \frac{1}{n}$ , regardless of what the $n$ values actually are or how they are spaced. The spec's own guidance for this item is a single short parenthetical: "the mean and variance of this distribution." Two points are worth being precise about, both directly checkable against the verified facts bank:

"Uniform" here means equally likely, not evenly sized or evenly spaced in a particular way. A real, verified WST01 question uses a discrete uniform distribution on the non-consecutive odd values $\{1,3,5,7\}$ for exactly this reason — the spacing is irrelevant to the definition (WST01-verified-facts.md §5, Jan 2025 Q1).
For a symmetric, evenly-spaced discrete uniform distribution specifically, the mean has a genuine shortcut: it sits at the midpoint of the smallest and largest value, $E(X) = \frac{1}{2}(x_{\min} + x_{\max})$ , without needing to sum every $x\,p(x)$ term individually. No equivalent shortcut for the variance is confirmed anywhere in the reviewed record — every real variance calculation on a discrete uniform variable found in the archive sums $x_i^2 p(x_i)$ term by term and applies $\mathrm{Var}(X) = E(X^2) - [E(X)]^2$ in full (WST01-verified-facts.md §5, §7; independently cross-checked against this same flag already recorded in codex-formula-and-method-bank.md §1.4 of this folder). A remembered $\frac{n^2-1}{12}$ -style closed form for the variance of a discrete uniform variable is not confirmed as examined content on this paper and should not be reached for as a shortcut here — the exam has, in the one clean worked example the facts bank found, explicitly asked for the variance "showing your working," which a memorised closed form would not demonstrate.

#5. Formula-booklet boundary for this topic, stated once, plainly

Directly from the verified facts bank's own reading of the Mathematical Formulae and Statistical Tables booklet (WST01-verified-facts.md §2a):

In the booklet (look up)	Not in the booklet (memorise)
$E(X) = \sum x_i P(X=x_i)$	$F(x_0) = P(X \le x_0) = \sum p(x)$ for $x \le x_0$
$\mathrm{Var}(X) = \sum x_i^2 P(X=x_i) - \mu^2$	—
$E(g(X)) = \sum g(x_i)P(X=x_i)$	—

The probability function's own definition, $p(x) = P(X=x)$ , and the two conditions in §2 are not formulae to look up at all — they are the definitional bedrock the rest of the topic sits on, and the facts bank's own §1.4 formula table (in codex-formula-and-method-bank.md) correctly labels them "technique, not a formula" for exactly this reason.

#6. Sub-case micro-examples

The four numbered items below are short, constructed illustrations of each real sub-case this spec point requires — none of these numbers come from a real past paper; they exist only to show each mechanism in isolation, in a couple of lines, and are clearly separate from the real-exam-angle synthesis in §7. The one real worked exemplar this lesson already carries in full (the Jan 2025 two-dice question) is deliberately not reproduced or re-derived here — see §8.

6.1 — Table-given, checking validity. $X$ takes values $2, 4, 6$ with $p(2)=0.3$ , $p(4)=0.5$ , $p(6)=0.2$ . Check: each value lies in $[0,1]$ , and $0.3+0.5+0.2=1$ — a valid probability function. Now suppose the same table is given but $p(6)$ is missing and only $p(2)=0.3$ , $p(4)=0.5$ are stated: the temptation is to treat the table as already complete. Summing what is given, $0.3+0.5=0.8 \ne 1$ , is the signal a value is missing, exactly the self-check §2.1's derivation justifies — here $p(6)=0.2$ is the forced remainder.

6.2 — Formula-given, unknown constant. $Y$ has probability function $p(y) = k(5-y)$ for $y=1,2,3,4$ . First line of working: $\sum p(y)=1$ , i.e. $k(4)+k(3)+k(2)+k(1) = 1 \Rightarrow 10k = 1 \Rightarrow k = \frac{1}{10}$ . (Written down as its own line, not skipped — see §7.2.)

6.3 — CDF construction and inversion. $X$ takes $0,1,2,3$ with $p(0)=0.1$ , $p(1)=0.3$ , $p(2)=0.4$ , $p(3)=0.2$ . Forward: $F(0)=0.1$ , $F(1)=0.4$ , $F(2)=0.8$ , $F(3)=1$ . Backward (§3.1): suppose instead only the cumulative table $F(0)=0.1$ , $F(1)=0.4$ , $F(2)=0.8$ , $F(3)=1$ had been given, with $p(2)$ left blank — it is recovered as $p(2) = F(2)-F(1) = 0.8-0.4 = 0.4$ , matching the value above without ever needing the $p(x)$ table directly.

6.4 — Complement rule applied to a CDF. Using the same $X$ as 6.3: $P(X>1) = 1-F(1) = 1-0.4=0.6$ — the ordinary complement rule (spec 3.2, $P(A')=1-P(A)$ ), applied to the event " $X\le1$ " rather than to a single value, and useful whenever a CDF table is given but the question asks for a "greater than" probability rather than a "less than or equal to" one.

6.5 — Discrete uniform on non-consecutive values. $Y$ is Discrete Uniform on $\{2,5,8,11\}$ — four equally likely, evenly-spaced but non-consecutive values, so $p(y)=\frac14$ each. By symmetry, $E(Y) = \frac{2+11}{2} = 6.5$ . No shortcut exists for the variance (§4): $E(Y^2) = \frac{2^2+5^2+8^2+11^2}{4} = \frac{4+25+64+121}{4} = \frac{214}{4} = 53.5$ , so $\mathrm{Var}(Y) = 53.5 - 6.5^2 = 53.5 - 42.25 = 11.25$ , found by full summation, term by term.

#7. Every real exam angle this specific topic has actually been tested from

Grounded entirely in WST01-verified-facts.md §3, §5 and §7, and the SIGNAL frequency document in this same folder. The methodology note those two documents already state applies here too: 6 of the 14 verified mark-scheme series (Jun 2019, Jan 2021, Oct 2021, Jun 2022, Jan 2023, Jun 2024) have a fully read examiner report; any "seen in N of M" figure below uses that 6 as its denominator unless stated otherwise. No likelihood percentage and no grade-boundary figure is stated anywhere in this section, or anywhere in this document — none exists in the verified record for this paper (signal-frequency-and-boundaries.md §2 confirms a direct search of the facts bank for grade-boundary terms returns no matches).

7.1 — A genuinely possible domain value left out entirely. The single most-quoted error on this spec point: "the most common error was to miss out the value x = 0" (Pearson examiner report, Oct 2021, Q4(d)) — a probability function whose stated values summed short of 1 because a real, valid value of $x$ (here, zero) was never written down at all, not because of an arithmetic slip in the values that were. §2.1's derivation is the direct defence: the sum falling short of 1 is the diagnostic signal.

7.2 — The $\sum p(x)=1$ equation itself omitted when solving for an unknown constant. The construction-side twin of 7.1, independently confirmed: candidates "failed to write down the equation for the sum of probabilities" (Pearson examiner report, Jun 2022, Q5(d)) even while going on to earn a later method mark — meaning the equation $\sum p(x)=1$ is itself a mark-earning line, separate from the algebra that solves it, on a "show that"-style item where the target constant is already printed on the page.

7.3 — The discrete uniform distribution appearing as an opening sub-part, not a standalone question. The facts bank flags this plainly as a genuine limitation of the verified record, not a gap in the research itself: beyond one clean worked example, the discrete uniform distribution "tends to appear as the opening, easiest part of a larger discrete-RV question" rather than anchoring a multi-part question of its own (WST01-verified-facts.md §5). The one clean example is Jan 2025 Q1(a), where naming a distribution as "Discrete uniform" earned a single mark with no calculation attached.

7.4 — The CDF's $\le$ / $<$ boundary and summation direction, tested implicitly by the formula not being printed. No examiner-report quote names this convention directly as a source of lost marks in the reviewed record, so this is stated as a structural fact rather than a confirmed error pattern: because $F(x_0)=P(X\le x_0)=\sum p(x)$ for $x\le x_0$ is explicitly excluded from the formula booklet (WST01-verified-facts.md §2a), any question using the CDF is, by construction, testing whether this definition — inequality direction included — was correctly recalled rather than looked up.

7.5 — One real question spanning almost the entire sub-topic in a single multi-part item. Jan 2025 Q1 (verified from the mark scheme) uses two four-sided dice to test, in one question: naming a discrete uniform distribution (a), a probability from that distribution (b), a mean read off "by symmetry" with no method credited by the mark scheme despite the question only saying "write down the value of" (c), a variance found "showing your working" with no shortcut available (d), two probabilities requiring full enumeration of a joint outcome space (e, f), and a missing CDF-table entry for a derived random variable recovered via the inversion technique in §3.1 (g) — a single question touching spec items 5.1, 5.2, 5.4 and, for the derived variable's construction, 3.1, together. The full working for this question is already reproduced stage-by-stage in the existing lesson file (see §8) and is not repeated here.

7.6 — Frequency across the checked record. Section 5 (discrete random variables as a whole) is confirmed tested, with a specific question or mark-scheme citation, in 5 of the 6 series with a fully read examiner report — Jan 2021, Oct 2021, Jun 2022, Jun 2019 and Jun 2024 — plus one further series, Jan 2025, confirmed present via a full mark-scheme worked example even though that series' own examiner report was not located (signal-frequency-and-boundaries.md §1). This is a lower bound on real frequency, not an exhaustive count: the facts bank's own §8 records that examiner reports for 8 of the 14 mark-scheme series were not located in the time available, so a topic's true appearance rate across the full 14-series archive is very likely higher than "5 of 6" understates it to be, not lower.

#8. Where the fuller worked treatment already lives (not duplicated here)

The full, stage-by-stage working of the Jan 2025 two-dice question (7.5 above) — all seven parts, including the CDF-inversion step for part (g) — lives in the worked chain titled "Two four-sided dice, from the real thing" inside content/veridian/curriculum/wst01/discrete-random-variables-probability-function.ts.
The June 2019 spinner question testing $E(X)$ against $P(\text{win})$ — a spec 5.3 point building on this topic's own $E(X)$ machinery — is fully worked in the same lesson file's second worked chain, and again as an independent RealPastQuestion entry in content/veridian/question-bank/wst01.ts.
The formula-booklet split in §5 above, and the "no confirmed variance shortcut for the discrete uniform distribution" flag in §4, are both already stated in codex-formula-and-method-bank.md §1.4 and §2.5 of this same folder — restated here only as far as needed for this document to stand alone, not duplicated in full.
A short, VERIDIAN-original "show that $k=\frac{1}{14}$ " marked solution, built around the exact 7.2 trap, already exists in the lesson file's own marked-solution block.

#9. Flagged limitations

Examiner reports for 8 of the 14 verified WST01 mark-scheme series were not located in the time available for the underlying research pass (Jun 2021, Jan 2022, Oct 2022, Jun 2023, Oct 2023, Jan 2024, Oct 2024, Jan 2025) — the exam-angle synthesis in §7 should be read as a lower bound on this topic's real testing history, not as an exhaustive account of it.
No confirmed closed-form variance shortcut exists in the reviewed record for a general discrete uniform distribution (§4) — this document does not supply one, since doing so would not be grounded in any verified source.
No grade-boundary data for WST01 exists anywhere in the sources this document or its companions were built from; none is stated here, consistent with the hard rule against inventing one.
The discrete uniform distribution specifically (spec 5.4) remains thin as a standalone topic in the verified record — one clean worked example, functioning as an opening sub-part of a larger question, is the only confirmed instance (§7.3).