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.
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.
A discrete random variable X takes countable values x1,x2,… with probabilities P(X=xi).
Expectation: E(X)=∑xiP(X=xi).
Second moment: E(X2)=∑xi2P(X=xi)
Variance: Var(X)=E(X2)−[E(X)]2.
Standard deviation: square root of Var(X).
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∼Bin(n,p) when you have n independent Bernoulli trials with success probability p.
Probability mass: P(X=r)=(rn)pr(1−p)n−r.
Expectation: E(X)=np.
Variance: Var(X)=np(1−p).
Conditions: fixed n, constant p, independent trials, success/failure outcomes.
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(X=4)=(412)0.340.78≈0.231.
P(X≥2)=1−[P(X=0)+P(X=1)]≈0.915
E(X)=12×0.3=3.6.
Var(X)=12×0.3×0.7=2.52.
Binomial-condition failure checkpoint
Before declaring X∼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 fails
Multiply the stage probabilities that match each round.
Averaging the probabilities and using that average as p.
More than two outcome types are possible
Success/failure setup needs defining
Define success clearly, then combine all other outcomes as failure only if the question allows it.
Counting only one failure type and losing probability mass.
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" or "no more than r"
P(X≤r)
Use cumulative probability up to r.
"At least r"
P(X≥r)
Use 1−P(X≤r−1)
"More than r"
P(X>r)
Use 1−P(X≤r)
"Fewer than r"
P(X<r)
Use P(X≤r−1)
Misconception check: the complement boundary moves by one when the inequality is inclusive. For P(X≥3), subtract P(X≤2), not P(X≤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 ∑P(X=x)=1.
A PMF that does not total 1 cannot be used for expectation or variance.
2
Find E(X)=∑xP(X=x).
This gives the balance point of the distribution.
3
Find E(X2)=∑x2P(X=x).
Variance needs the second moment, not just the mean.
4
Subtract [E(X)]2 from E(X2).
This avoids the common error Var(X)=E(X2)−E(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.
Several entries in terms of k.
Add every expression in the probability row, then solve ∑P(X=x)=1.
Each probability expression must be non-negative after substitution.
An extra condition, such as E(X)=c.
Use the total-probability equation first, then use the mean condition if one unknown remains.
Do not use E(X) until the probability row is a valid PMF.
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=307. Only after this check should you compute E(X):
E(X)=0(0.2)+1(307)+2(3014)+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(X=x)
0.1
0.3
0.4
k
Use the total probability condition to find k=0.2.
E(X)=0×0.1+1×0.3+2×0.4+3×0.2=1.7.
E(X2)=02×0.1+12×0.3+4×0.4+9×0.2=3.7
Var(X)=3.7−1.72=0.81.
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
Forgetting to add the fixed shift b.
Variance
Var(Y)=a2Var(X)
Adding b, or multiplying by a
Standard deviation
SD(Y)=∣a∣SD(X)
Squaring the standard deviation multiplier.
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)=aE(X)+b and Var(Y)=a2Var(X).
Linear combinations: for independent X and Y, Var(X+Y)=Var(X)+Var(Y)
Example -- Bonus payouts
Let X be binomial Bin(8,0.4). Each success triggers a 200 dollar bonus plus a fixed 150 dollar stipend. Total payout T=200X+150.
E(T)=200×8×0.4+150=790 dollars.
Var(T)=2002×8×0.4×0.6=76800 dollars squared.
Standard deviation ≈277 dollars.
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)
Var(T)=Var(X)+Var(Y)
Subtracting or squaring means instead of adding variances.
T=X−Y, with independent X and Y
E(T)=E(X)−E(Y)
T=3X
E(T)=3E(X)
Var(T)=9Var(X)
T=X1+X2+X3
Worked check: if E(X)=4, Var(X)=2, E(Y)=7, and Var(Y)=5, then for independent X and Y, E(X−Y)=−3 but Var(X−Y)=7. The mean can be negative; the variance cannot.
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∼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 Var(X) as E(X2)−E(X) instead of E(X2)−[E(X)]2: Forgetting to square E(X) in the shortcut formula is one of the most common arithmetic errors. Write the formula in full before substituting.
Scaling variance incorrectly in linear transformations: For Y=aX+b, Var(Y)=a2Var(X) - the constant
Forgetting to check that probabilities sum to 1: In custom PMF questions, always verify ∑P(X=xi)=1 before computing expectation. Missing this step means you may use an invalid probability distribution.
Using cumulative probability incorrectly for "at least" or "at most" problems: For P(X≥k), use the complement 1−P(X≤k−1) rather than 1−P(X≤k)
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 (rn)pr(1−p)n−r and the binomial mean np and variance np(1−p), but the general E(X)=∑xiP(X=xi) definition, the shortcut Var(X)=E(X2)−[E(X)]2, the linear-transformation rules, and all four binomial conditions must be recalled from memory.
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∼Bin(n,p), state the probability expression (e.g. P(X≥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 (rn)pr(1−p)n−r or the GC.
Other H2 Maths formula sheets
Revising more than one topic? Grab the matching one-page formula sheet:
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
