Unit 5 · Probability & Statistics 2 (Paper 6, A Level)
The Complete Cambridge International AS & A Level Mathematics (9709) Guide · Cambridge International AS & A Level Mathematics (9709)Unit 5 · Probability & Statistics 2 (Paper 6, A Level)
- The Complete Cambridge International AS & A Level Mathematics (9709) Guide
- Cambridge International AS & A Level Mathematics (9709)
- 5 sections
Cambridge International does not publish a per-unit weighting for this exam. Exam structure (official): AS Level: Pure Mathematics 1 (110 min, 60%) · Pure Mathematics 2 (75 min, 40%) · Mechanics (75 min, 40%) · Probability & Statistics 1 (75 min, 40%); A Level: Pure Mathematics 1 (110 min, 30%) · Pure Mathematics 3 (110 min, 30%) · Mechanics (75 min, 20%) · Probability & Statistics 1 (75 min, 20%) · Probability & Statistics 2 (75 min, 20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. A complete conclusion includes parameters, assumptions, numerical evidence, and context.
- How Cambridge International AS & A Level Mathematics (9709) assesses this Cambridge International publishes no per-unit percentage for Cambridge International A Level Mathematics; Pure Mathematics 1 accounts for 60% of the AS Level total and 30% of the A Level total. · AS Level: Pure Mathematics 1 (110 min, 60%) · Pure Mathematics 2 (75 min, 40%) · Mechanics (75 min, 40%) · Probability & Statistics 1 (75 min, 40%) | A Level: Pure Mathematics 1 (110 min, 30%) · Pure Mathematics 3 (110 min, 30%) · Mechanics (75 min, 20%) · Probability & Statistics 1 (75 min, 20%) · Probability & Statistics 2 (75 min, 20%) · show the model with Poisson probability statement with a rescaled mean, continuous density graph or integral over its support, confidence interval and standardized test statistic with a contextual decision
- Key skills The Poisson Distribution and Its Approximations, Linear Combinations of Random Variables, Continuous Random Variables: Probability Density Functions, Mean, Variance and Percentiles, Sampling and Estimation: Central Limit Theorem, Unbiased Estimates and Confidence Intervals, Hypothesis Tests: Binomial, Poisson and Normal Means, Type I and Type II Errors
- How to study for Unit 5 This page turns Poisson probability statement with a rescaled mean, continuous density graph or integral over its support, confidence interval and standardized test statistic with a contextual decision into one route: Write the model, exposure or support, and the parameter or hypothesis being estimated or tested..
- The organizing decision Choose the probability model or inference procedure, state its parameters and assumptions, and connect the statistic to a contextual conclusion.
What Cambridge International AS & A Level Mathematics (9709) Unit 5 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
The Poisson Distribution and Its Approximations
Poisson mean; exposure; linear combination02Linear Combinations of Random Variables
linear combination; variance; density function03Continuous Random Variables: Probability Density Functions, Mean, Variance and Percentiles
density function; support; standard error04Sampling and Estimation: Central Limit Theorem, Unbiased Estimates and Confidence Intervals
standard error; confidence interval; p-value05Hypothesis Tests: Binomial, Poisson and Normal Means, Type I and Type II Errors
p-value; Poisson mean; exposureCambridge International does not publish a per-unit weighting for this exam.
Official unit name and weighting: Cambridge International syllabus.
Modeling and inference
Connect the published share to the unit model
A complete conclusion includes parameters, assumptions, numerical evidence, and context.
For modeling and inference, the following pages use newly written examples and data. The modeling and inference scoring comparisons are learning aids based on the published syllabus and the general requirement to show necessary working, not official mark schemes.
Before work on modeling and inference, identify the domain, units, accuracy instruction, and event boundary. Preserve exact values in modeling and inference until a final approximation is requested.
The decision that organizes this unit
Define the system and choose the route before calculating
Choose the probability model or inference procedure, state its parameters and assumptions, and connect the statistic to a contextual conclusion.
Write the model, exposure or support, and the parameter or hypothesis being estimated or tested.
Mechanism route and repair branches
Relationships to preserve
- A Poisson mean scales with exposure before point or tail probabilities are evaluated.
- Means transform linearly, while independent variances use squared coefficients.
- A confidence interval quantifies plausible values, while a hypothesis test compares evidence with a stated significance level.
Representations to read
- Poisson probability statement with a rescaled mean
- continuous density graph or integral over its support
- confidence interval and standardized test statistic with a contextual decision
Branches to reject
- using an unscaled rate for a different exposure
- treating a density height as a point probability or integrating outside the support
- using the population standard deviation as the standard error or claiming that a test proves a hypothesis
| Key concept | Why it's hard | What scores |
|---|---|---|
| Poisson and combinations | Exposure and squared coefficients are missed | Rescales rates and transforms variance |
| Continuous variables | Density height is not point probability | Integrates over the requested support |
| Inference | A decision is not proof | States hypotheses, uncertainty, and evidence |
How Cambridge International AS & A Level Mathematics (9709) assesses Probability & Statistics 2 (Paper 6, A Level)
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| The Poisson Distribution and Its Approximations | Poisson mean; exposure; linear combination | using an unscaled rate for a different exposure |
| Linear Combinations of Random Variables | linear combination; variance; density function | treating a density height as a point probability or integrating outside the support |
| Continuous Random Variables: Probability Density Functions, Mean, Variance and Percentiles | density function; support; standard error | using the population standard deviation as the standard error or claiming that a test proves a hypothesis |
| Sampling and Estimation: Central Limit Theorem, Unbiased Estimates and Confidence Intervals | standard error; confidence interval; p-value | using an unscaled rate for a different exposure |
| Hypothesis Tests: Binomial, Poisson and Normal Means, Type I and Type II Errors | p-value; Poisson mean; exposure | treating a density height as a point probability or integrating outside the support |
Poisson rates
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Poisson rates
Calls arrive at a help desk at a mean rate of 3.6 per hour.
Using a Poisson model, find (a) the probability of exactly two calls in a period of 20 minutes, and (b) the probability of at least one call in that period.
[4]Show all necessary working.
A. Use λ = 1.2; calculate e⁻¹·²(1.2²)/2! and 1 − e⁻¹·².
✅ Correct! 😊 Correct. Twenty minutes is one third of an hour, and the complement of no calls gives the probability of at least one.
B. Use λ = 3.6; calculate e⁻³·⁶(3.6²)/2! and 1 − e⁻³·⁶.
❌ Not quite 😭 This uses the hourly mean for a 20-minute exposure. The mean must be rescaled before either probability is evaluated.
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C. Use λ = 1.2; calculate e⁻¹·²(1.2²) and e⁻¹·².
❌ Not quite 😭 The first expression omits 2!, and the second gives no calls rather than at least one call.
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D. Use λ = 0.6; calculate e⁻⁰·⁶(0.6²)/2! and 1 − e⁻⁰·⁶.
❌ Not quite 😭 This halves the hourly mean instead of multiplying it by 20/60. It therefore models the wrong exposure length.
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Full explanation
- Step 1Rescale the hourly mean to 20 minutes: λ = 3.6 × 20/60 = 1.2.
- Step 2For X ∼ Poisson(1.2), P(X = 2) = e⁻¹·²(1.2²)/2! ≈ 0.217.
- Step 3Use the complement for at least one call: P(X ≥ 1) = 1 − P(X = 0).
- Step 4Thus P(X ≥ 1) = 1 − e⁻¹·² ≈ 0.699.
Practice 2: Continuous probability density
A continuous random variable X has probability density function f.
The density is given by f(x) = kx for 0 ≤ x ≤ 2, and f(x) = 0 otherwise. Find k and P(X > 1).
[5]Show all necessary working.
A. Set ∫ from 0 to 2 of kx equal to 1, then integrate kx from 1 to 2.
✅ Correct! 😊 Correct. Total area over the support determines k, and probability over an interval is the corresponding area under the density.
B. Set f(2) = 1, then use P(X > 1) = f(1).
❌ Not quite 😭 A density need not have height 1, and its value at a single point is not the probability of an interval.
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C. Set ∫ from 0 to 1 of kx equal to 1, then subtract f(1) from 1.
❌ Not quite 😭 This normalises over only part of the support and then treats a density height as a cumulative probability.
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D. Set ∫ from −∞ to ∞ of kx equal to 1, using kx outside 0 ≤ x ≤ 2.
❌ Not quite 😭 The formula kx applies only on the stated support. Extending it outside that interval contradicts the piecewise definition.
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Full explanation
- Step 1Normalisation gives ∫ from 0 to 2 of kx = 1.
- Step 2Evaluating gives 2k = 1, so k = 1/2.
- Step 3Then P(X > 1) = ∫ from 1 to 2 of x/2.
- Step 4The integral is [x²/4] from 1 to 2 = 3/4.
Key terms for Unit 5: Probability & Statistics 2 (Paper 6, A Level)
Models, uses, and boundaries
- The inference sequence moves from a declared model to a statistic and then to evidence
- Scale the Poisson mean to the stated time, then evaluate the exact count and complement events. The inference sequence moves from a declared model to a statistic and then to evidence.
- The Poisson mean scales with time before the point probability is evaluated
- Transform the mean with the original coefficients and transform independent variances with squared coefficients. The Poisson mean scales with time before the point probability is evaluated.
- Linearity gives the combined mean, and independence gives the combined variance
- Normalize the density over its support, integrate the requested interval, and integrate value times density for the expectation. Linearity gives the combined mean, and independence gives the combined variance.
- Normalisation determines the density constant before the required interval is integrated
- Use the standard error to build the confidence interval and standardize the sample mean for the one-sided test. Normalisation determines the density constant before the required interval is integrated.
- The confidence interval and standardised test statistic display the estimation and decision calculations
- Move from a declared model to a statistic, then state what the numerical evidence supports in context. The confidence interval and standardised test statistic display the estimation and decision calculations.
Cambridge International AS & A Level Mathematics (9709) Unit 5 FAQ
How much of Cambridge International AS & A Level Mathematics (9709) does Unit 5 carry?
Cambridge International does not publish a per-unit weighting for this exam.
What is the first move on a Probability & Statistics 2 (Paper 6, A Level) problem?
Write the model, exposure or support, and the parameter or hypothesis being estimated or tested.
Which relationships should I preserve?
A Poisson mean scales with exposure before point or tail probabilities are evaluated. Means transform linearly, while independent variances use squared coefficients. A confidence interval quantifies plausible values, while a hypothesis test compares evidence with a stated significance level.
Which representations should I practice?
Practice moving among Poisson probability statement with a rescaled mean, continuous density graph or integral over its support, confidence interval and standardized test statistic with a contextual decision.
What error should I check before submitting an answer?
Check for using an unscaled rate for a different exposure; treating a density height as a point probability or integrating outside the support; using the population standard deviation as the standard error or claiming that a test proves a hypothesis.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Poisson rates
- Linear combinations
- Continuous density
- Intervals and tests
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related Cambridge International AS & A Level Mathematics (9709) unit guides
Cambridge International AS & A Level Mathematics (9709) Exam Guide & Review
The whole exam and its official unit sequence.01Pure Mathematics 1 (Paper 1)
Cambridge International does not publish a per-unit weighting for this exam.02Pure Mathematics 3 (Paper 3, A Level)
Cambridge International does not publish a per-unit weighting for this exam.03Mechanics (Paper 4)
Cambridge International does not publish a per-unit weighting for this exam.04Probability & Statistics 1 (Paper 5)
Cambridge International does not publish a per-unit weighting for this exam.How to study Cambridge International AS & A Level Mathematics (9709) Unit 5
Start with the organizing decision
Before solving, restate the decision in operational terms: Choose the probability model or inference procedure, state its parameters and assumptions, and connect the statistic to a contextual conclusion.. Your first written move should be to Write the model, exposure or support, and the parameter or hypothesis being estimated or tested..
Practice the same idea in several representations
Rotate through Poisson probability statement with a rescaled mean, continuous density graph or integral over its support, confidence interval and standardized test statistic with a contextual decision. Use each representation to practice The Poisson Distribution and Its Approximations, Linear Combinations of Random Variables, Continuous Random Variables: Probability Density Functions, Mean, Variance and Percentiles, Sampling and Estimation: Central Limit Theorem, Unbiased Estimates and Confidence Intervals, Hypothesis Tests: Binomial, Poisson and Normal Means, Type I and Type II Errors, and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for using an unscaled rate for a different exposure; treating a density height as a point probability or integrating outside the support; using the population standard deviation as the standard error or claiming that a test proves a hypothesis. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official Cambridge International syllabus for the 2026-2027 administration.
Where to go next
Return to the Cambridge International AS & A Level Mathematics (9709) to see every unit and plan your revision.