Match a Poisson count with a suitable normal approximation
Check the large-parameter guide and match mean and variance before approximating integer probabilities with continuous areas.
Section 6.1 · The Poisson distribution
Exam scope and assessment notes
Probability & Statistics 2 (Paper 6, A Level) · Mathematics 9709, 2026–2027, section 6.1. Paper 6 builds on Paper 5 and assumes Paper 3 calculus. Po(λ) counts non-negative integers over a stated exposure, with finite λ ≥ 0. Mean and variance are each λ; proofs are not required. The homogeneous process examples stipulate constant rate, independent counts in disjoint exposures, and negligible multiple events in sufficiently small exposures; real applications need evidence. A binomial-to-Poisson approximation uses large n and small p, with n > 50 and np < 5 approximately as the syllabus guide. A Poisson-to-normal approximation uses λ > 15 approximately, matching N(λ,λ) and applying continuity correction. These guides are not universal error guarantees. Exact and approximate calculations are labelled separately; numerical normal probabilities use unrounded z and can differ slightly from rounded-table working. Independent Poisson sums and linear combinations, sample means and hypothesis testing belong to later Statistics 2 sections. Written practice is self-checked; cumulative assessment and human subject-teacher review remain pending.
Before you begin
- Use Poisson moments.
- Read normal variance and standard deviation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which normal variance matches Po(20)?
What you will learn
- Check λ > 15 approximately as the stated guide.
- Match N(λ,λ), with SD √λ.
Match the moments
Cambridge gives λ > 15 approximately for a normal approximation to Po(λ). For Po(20), use Y ~ N(20,20), SD √20. The second parameter is variance, not SD twenty or variance four hundred. X remains discrete and Y is a continuous approximation.
Po(15) does not meet the stated greater-than-fifteen guide. This practical criterion does not prove every calculation at fifteen is unusable. Exact Poisson calculation remains available. Suitable normal approximations still need continuity correction for integer events.
Exposure changes suitability
At ten per hour, two hours gives λ = 20 and meets the guide, while half an hour gives λ = 5 and does not. Check the observation’s expected count, rather than using rate directly as the parameter.
Normal models assign some probability to negative real values, whereas Poisson counts cannot be negative. The models therefore remain different. The diagram compares probability-cell areas with normal density on a common standardised scale, showing only part of both ranges.
Pause and explain
At ten per hour, which exposure meets the stated normal guide?
Worked example
Choose a matching normal approximation to Po(20) and state SD.
Show the worked solution
- Twenty exceeds the approximate guide fifteen.
- Use mean and variance twenty: N(20,20).
- SD is √20; integer boundaries need correction.
Answer Approximately N(20,20), SD √20.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →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.
Check and state a normal approximation for Po(20), including SD.
Give me a hint
Match both moments after checking the parameter guide.
Compare my reasoning
- 20 > 15 meets the guide.
- Use N(20,20).
- SD = √20 ≈ 4.4721; correct the count-event boundaries.
Approximately N(20,20), SD √20.
Look for these in your work
- State suitability as a guide.
- Distinguish variance and SD.
Does Po(15) meet the stated guide? What gives exact model probabilities?
Give me a hint
Read the inequality without claiming every approximation impossible.
Compare my reasoning
- The guide is λ > 15 approximately.
- Fifteen does not exceed fifteen.
- Exact masses or sums give the Poisson model probabilities.
Guide not met; exact Poisson calculation remains available.
Look for these in your work
- Check the written guide.
- Avoid overclaiming numerical accuracy.
At ten per hour, compare two hours and half an hour for normal suitability.
Give me a hint
Find both parameters before checking the guide.
Compare my reasoning
- Two hours gives λ = 20.
- Half an hour gives λ = 5.
- First matches N(20,20) by guide; second does not meet it.
Two hours: guide met; half-hour: guide not met.
Look for these in your work
- Scale both exposures.
- Check the modelled count.
Common mistakes
- Check the actual exposure parameter.
- Normal second parameter is variance.
- Suitability does not remove continuity correction.
What can you explain now?
Does Po(15) meet the written guide?
Compare with the explanation
No; fifteen is not greater than fifteen.
The guide is approximate rather than a universal error guarantee.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Find exposure-specific λ, check suitability, match moments and then preserve integer cells.
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.