H2 Maths Discrete Random Variables Formula Sheet

Study guide07 Oct 2025, 00:00 ZUpdated 17 Jul 2026

H2 Maths discrete random variables formula sheet: probability distributions, expectation E(X), variance Var(X), the binomial distribution and its four conditions, linear transfo...

Q: What does H2 Maths Notes (JC 1-2): 6.2) Discrete Random Variables cover?
A: Expectation, variance, binomial modelling, and calculator workflows for H2 discrete distributions.

Download: Get the H2 Maths Discrete Random Variables formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.

Study cadence
Re-derive the expectation and variance formulas from first principles once per week so you remember why each term appears. Keep a small table template in your notes for probability mass functions (PMFs) so you can slot in values quickly during exams.

A discrete random variable counts possible outcomes: List the values it can take.
Expectation is the long-run average: Multiply each value by its probability.
Binomial questions need fixed trials, constant probability, independence, and success/failure outcomes: Check the four conditions before using the formula.

Concrete example: If 12 candidates each independently accept with probability 0.3, the number who accept is binomial. If one acceptance affects another, it is not.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 6.2 is assessed in Paper 2 Section B (Probability and Statistics, 60 marks) and focuses on discrete distributions and the binomial model.

Formulas at a glance

Every result the 9758 syllabus expects you to recall, on one screen. MF27 supplies the binomial probability mass formula and its mean and variance, but you still need to recall the general expectation and variance definitions, linear-transformation rules, and binomial conditions. Worked examples for each appear in the sections below.

Expectation and variance

Quantity	Formula
Expectation	$E(X) = \sum x_i P(X = x_i)$

Linear transformations

Quantity	Formula
Mean of $Y = aX + b$	$E(Y) = aE(X) + b$

Binomial distribution

Quantity	Formula
Notation	$X \sim \operatorname{Bin}(n, p)$
Probability mass	$P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$

Core Concepts

A discrete random variable $X$ takes countable values $x_1, x_2, \dots$ with probabilities $P\left(X = x_{\mathrm{i}}\right)$

The probability distribution of a discrete random variable lists each value x with its probability P(X = x); the probabilities are non-negative and sum to 1. The expectation E(X) is the balance point Σ x·P(X = x), and the variance measures the spread about it.

Binomial Model

Use $X \sim \operatorname{Bin}(n, p)$ when you have $n$ independent Bernoulli trials with success probability $p$ .

Probability mass: $P\left( X = r \right) = \binom{n}{r} p^r (1 - p)^{n - r}$

Example -- Hiring round

A firm interviews $n = 12$ candidates. Each has a 0.3 probability of accepting an offer. Let $X$ be the number who accept.

$P\left( X = 4 \right) = \binom{12}{4} 0.3^4 0.7^8 \approx 0.231$

Binomial-condition failure checkpoint

Before declaring $X \sim \operatorname{Bin}(n,p)$ , test the wording against all four binomial conditions. If one condition fails, write the reason instead of forcing the formula.

Wording clue	Condition at risk	What to write instead	Common trap
"Without replacement" from a small group	Independence and constant $p$ may fail	Use a tree diagram or hypergeometric-style counting from the changing group.	Keeping the same probability on every branch.
"Until the first success"	Fixed number of trials fails	Model the stopping rule directly instead of using a fixed $n$ .	Treating the number of trials as known before the experiment starts.
Success probability changes by round	Constant $p$

Worked check: drawing $3$ students without replacement from a group of $8$ boys and $7$ girls is not automatically binomial. After one boy is drawn, the probability of another boy changes from $8/15$ to $7/14$ , so independence and constant $p$ fail.

Misconception check: "counting successes" is not enough for a binomial model. The trials must also be fixed, independent, have constant success probability, and have only success/failure outcomes.

Binomial Tail Checkpoint

Translate the wording before opening the calculator menu.

Question wording	Probability to enter	Safer calculator route
"Exactly $r$ "	$P(X = r)$	Use the probability distribution value for $r$ .
"At most $r$

Misconception check: the complement boundary moves by one when the inequality is inclusive. For $P(X \geq 3)$ , subtract $P(X \leq 2)$ , not $P(X \leq 3)$ .

Custom PMFs

When the PMF is tabulated (common in H2 questions):

Use this order before touching the calculator:

Step	What to compute	Why it comes first
1	Check $\sum P(X = x) = 1$ .	A PMF that does not total 1 cannot be used for expectation or variance.
2	Find $E(X) = \sum xP(X = x)$

The sequence matters: normalise first, then compute the mean, then compute the second moment, then finish the variance.

Unknown-parameter PMF checkpoint

When a probability is written as $k$ , $ak$ , or $1 - k$ , find the parameter before calculating any mean or variance.

Clue in the table	First equation to write	Check before moving on
One unknown probability, such as $k$ .	Total the known probabilities and set the full sum equal to $1$ .	The final $k$ must lie between $0$ and $1$

Worked check: suppose $X$ takes values $0, 1, 2, 3$ with probabilities $0.2, k, 2k, 0.1$ . The total-probability equation is

$0.2 + k + 2k + 0.1 = 1.$

So $3k = 0.7$ , giving $k = \frac{7}{30}$ . Only after this check should you compute $E(X)$

$E(X) = 0(0.2) + 1\left(\frac{7}{30}\right) + 2\left(\frac{14}{30}\right) + 3(0.1).$

Misconception check: do not treat $k$ as the value of $X$ . It is part of the probability row, so it is fixed by total probability before the random variable calculations start.

$x$	$0$	$1$	$2$	$3$
$P\left( X = x \right)$

Use the total probability condition to find $k = 0.2$ .
$E(X) = 0 \times 0.1 + 1 \times 0.3 + 2 \times 0.4 + 3 \times 0.2 = 1.7$

Transformations

Before applying a linear transformation, decide whether the question asks for a mean, a variance, or a standard deviation. They do not scale in the same way.

Quantity asked for	If $Y = aX + b$ , use	Common trap
Mean	$E(Y) = aE(X) + b$

Misconception check: a fixed allowance, fee, or stipend shifts every outcome by the same amount. It changes the mean, but it does not change spread.

For $Y = aX + b$ : $E(Y) = a E(X) + b$ and $\operatorname{Var}(Y) = a^2 \operatorname{Var}(X)$

Example -- Bonus payouts

Let $X$ be binomial $\operatorname{Bin}(8, 0.4)$ . Each success triggers a 200 dollar bonus plus a fixed 150 dollar stipend. Total payout $T = 200 X + 150$ .

$E(T) = 200 \times 8 \times 0.4 + 150 = 790$ dollars.
$\operatorname{Var}(T) = 200^2 \times 8 \times 0.4 \times 0.6 = 76800$

Independent-sum variance checkpoint

When a question combines two random variables, separate the mean calculation from the spread calculation. The signs and constants do not behave the same way.

Situation	Mean rule	Variance rule	Trap to avoid
$T = X + Y$ , with independent $X$ and $Y$	$E(T) = E(X) + E(Y)$

Worked check: if $E(X) = 4$ , $\operatorname{Var}(X) = 2$ , $E(Y) = 7$ , and $\operatorname{Var}(Y) = 5$

Misconception check: independence is needed before adding variances. If the variables are linked, the simple variance-addition rule is not justified.

Calculator Workflows

Casio: BPD (binomial probability distribution) for individual terms, BCD (binomial cumulative distribution) for cumulative sums. Store n and p in variables for quick reuse.
TI: binompdf(n, p, r) and binomcdf(n, p, r) functions in the DISTR menu. For PMF tables, program a short list-based routine to multiply L1 (values) and L2 (probabilities).
Cross-check expectation and variance using calculator built-ins but still show the manual summation in working.

Exam Watch Points

State whether a binomial model is appropriate; include independence and constant probability statements.
When a question mentions "at least one" or "no more than", use complements to reduce calculator input.
For custom PMFs, ensure probabilities sum to 1 before computing expectation.
Payout or cost problems often require a linear transformation-write the transformation before substituting.

Practice Quiz

Check your understanding of discrete PMFs, expectation/variance, and binomial modelling decisions.

Quick Revision Checklist

Identify when binomial modelling fits and articulate the assumptions.
Compute expectation and variance efficiently, including from tables.
Execute linear transformations and interpret the resulting mean and variance.
Use calculator functions accurately while documenting the method in words.
Use complements to simplify “at least/at most” binomial probabilities.

Want weekly guided practice on Discrete Random Variables? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

Using the binomial model without checking the conditions: Jumping to $X \sim \operatorname{Bin}(n, p)$ without stating that trials are independent, $p$ is constant, and outcomes are success/failure will cost the "state conditions" mark. Always write out all four conditions.
Computing $\operatorname{Var}(X)$ as $E(X^2) - E(X)$

Frequently asked questions

Is there a formula sheet for H2 Maths discrete random variables?
Yes - the "Formulas at a glance" section near the top of this page collects every result you need: the general expectation and variance definitions, linear transformation rules, independent-sum variance, and the full binomial distribution. MF27 provides the binomial probability mass formula $\binom{n}{r}p^r(1-p)^{n-r}$

Is Topic 6.2 in Paper 1 or Paper 2?
Topic 6.2 is part of Probability & Statistics and is assessed in Paper 2 Section B (60 marks). Paper 1 covers Pure Mathematics only.

Can I use the GC for binomial probability calculations?
Yes. binompdf and binomcdf (TI) or BPD and BCD (Casio) are expected tools. You must still write down the model $X \sim \operatorname{Bin}(n, p)$ , state the probability expression (e.g. $P(X \geq 3)$ ), and show that you are using the complement or cumulative form correctly.

What is the difference between a discrete random variable question and a binomial question?
A general discrete random variable question gives you a custom PMF table, while a binomial question describes a scenario with $n$ independent trials. The binomial is a special case of a discrete random variable. For a custom PMF, you work directly from the table; for binomial, you use the formula $\binom{n}{r}p^r(1-p)^{n-r}$

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

Functions & graphs: Functions · Graphs & Transformations · Equations & Inequalities
Sequences & series: Sequences & Series
Vectors: Vectors
Calculus: Differentiation · Maclaurin Series · Integration Techniques · Definite Integrals · Differential Equations
Statistics: Probability · Discrete Random Variables (this page) · Normal Distribution · Sampling · Hypothesis Testing · Correlation & Regression

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 6 Probability and statistics sub-topic 6.2 Discrete random variables (expectation, variance, binomial distribution, linear transformations): SEAB H2 Mathematics syllabus PDF

Next steps: follow the H2 Maths notes hub into Topic 6.3 - Normal distribution for approximation workflows.

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Reviewed by

Marcus Pang·Managing Director (Maths)