A Level · Cambridge (CIE) · 9709

Use exponential density areas and check divergent moments

Integrate a single-interval exponential density and contrast its finite moments with a valid heavy-tail model.

Section 6.3 · Continuous random variables

Statistics 2 · Continuous random variables · 6.3

Exam scope and assessment notes

Probability & Statistics 2 (Paper 6, A Level) · Mathematics 9709, 2026–2027, section 6.3. Paper 6 builds on Paper 5 and assumes Paper 3 calculus. A probability density is non-negative and has total area one over a single interval; its support may be infinite. Density height is not probability, and individual points have probability zero. Calculate interval probabilities, median and other percentiles directly from density areas. Explicit cumulative distribution function knowledge is not required by this section. Calculate E(X) and E(X²) by integration, then Var(X) = E(X²) − E(X)² when the needed moments are finite. Infinite-support densities require convergent improper integrals; a valid density need not have finite moments. Only single-interval densities are used here, with zero density outside the stated support. Model conclusions are conditional on the stipulated distribution; examples do not establish a real population model. Sampling, estimation and hypothesis testing belong to later sections. Written practice is self-checked; cumulative assessment and human subject-teacher review remain pending.

Before you begin

Integrate exponentials and use integration by parts.
Evaluate logarithmic limits.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which integral checks the mean of density 1/x² on x ≥ 1?

What you will learn

Calculate exponential areas and moments with Paper 3 calculus.
Recognise a valid density whose mean diverges.

Density 2e^(−2x) on x ≥ 0 is drawn only to three. Shading shows the visible part of X > 1, whose complete tail continues indefinitely and has area e⁻². Mean 1/2 and variance 1/4 are finite. The median ln(2)/2 splits total area; plotted endpoints do not bound support.

An infinite interval can have finite moments

For f(x) = 2e^(−2x) on x ≥ 0, total area is one and P(X > t) = e^(−2t) for t ≥ 0. The median is ln(2)/2. A threshold containing 95% of values satisfies e^(−2t) = 0.05, so t = ln(20)/2 ≈ 1.4979.

Integration by parts gives E(X) = 1/2 and E(X²) = 1/2, hence variance 1/4. The exponential factor drives the boundary terms x e^(−2x) and x² e^(−2x) to zero. The formulae are derived for this stipulated density; recognising a named exponential distribution is not an extra syllabus requirement.

Unit area does not guarantee a finite mean

The density 1/x² on x ≥ 1 also has total area one. But its mean integrand is 1/x, whose integral from one to R is ln R and diverges. There is no finite mean or finite variance to obtain by ordinary moment subtraction.

This contrast is a convergence check for infinite-support densities, not a required classification of heavy-tail families. Do not subtract two divergent expressions or assign an artificial cutoff without stating a new truncated and re-normalised model.

Pause and explain

What follows when the first-moment integral diverges?

Put the idea to work

Worked example

For density 2e^(−2x), find P(X > 1) and its median.

Show the worked solution

Upper area at one is e⁻².
Set e^(−2m) = 1/2.
m = ln(2)/2.

Answer Tail e⁻² ≈ 0.1353; median ln(2)/2.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

For 2e^(−2x) on x ≥ 0, find P(X > 1) and median.

Give me a hint

Use tail area e^(−2t).

Compare my reasoning

Integrate 2e^(−2x) from one to infinity to obtain e⁻².
Median solves e^(−2m) = 1/2.
Taking logarithms gives m = ln(2)/2, a non-negative supported location.

Tail e⁻² ≈ 0.1353; median ln(2)/2.

Look for these in your work

Use an infinite-limit area.
Solve a half-area equation.

2 · Independent

Find mean, second moment, variance and a 95% upper threshold for this density.

Give me a hint

Use integration by parts and a 0.05 upper tail.

Compare my reasoning

Mean = 1/2; second moment = 1/2.
Variance = 1/4.
Threshold is ln(20)/2.

Mean 1/2; second moment 1/2; variance 1/4; threshold ≈ 1.4979.

Look for these in your work

Check moment limits.
Use the correct tail probability.

3 · Transfer

Check whether density 1/x² on x ≥ 1 has unit area and a finite mean.

Give me a hint

Normalisation and first moment use different powers.

Compare my reasoning

∫₁∞x⁻² dx = 1.
The mean is lim(R→∞) ln R.
It diverges, so no finite mean or finite variance is available.

Valid density with area 1, but divergent mean; no finite moment-based variance.

Look for these in your work

Separate validity from convergence.
Do not subtract divergent quantities.

Common mistakes

Do not substitute infinity into an unevaluated expression.
Check moment boundary terms.
Do not subtract divergent moments.

Recall without your notes

What can you explain now?

Does total density area one guarantee finite expectation?

Compare with the explanation

No.

The valid 1/x² density has a divergent mean integral.

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Make it stick

Check area, mean and second moment as separate improper integrals.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.