LearnYMA01WST02 · Statistics 2The binomial and Poisson distributionsScaling rates and sums

The binomial and Poisson distributions · Subtopic 2 of 3

Scaling Rates and Adding Poisson Variables

If events happen at $4$ per hour, they happen at $2$ per half-hour: scale $\lambda$ with the interval. And two independent Poisson streams combine into one, with the rates simply added.

Cambridge 9709 Paper 6Edexcel YMA01 WST02Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Scale $\lambda$ in proportion to the length of the interval.
Find $\lambda$ from a given probability, using logarithms.
Use $X+Y\sim\text{Po}(\lambda+\mu)$ for independent $X\sim\text{Po}(\lambda)$ , $Y\sim\text{Po}(\mu)$ .
Combine scaling and sums in multi-step questions.

Pause and think

A shop gets $3$ emails and $2$ texts an hour, on average, at random and independently. How would you model the total number of messages in an hour? In $3$ hours?

Section 2 of 14

Core idea

Start hereLonger wait, more calls

If calls average 2 per hour, then in 3 hours they average 6, and in half an hour 1. The rate scales with the length of the interval, and the Poisson parameter scales with it.

The big idea

Rate $\lambda$ per unit ⇒ count in $t$ units is $\operatorname{Po}(\lambda t)$ . Separate intervals are independent.

1. Scaling

Random events on a timeline: same rate, longer window, more events

Counts vary from interval to interval (3, 3, 1, 4 here), but on average each 15 minutes holds the same number.

Count in a window of t hours at 2 per hour: Po(2t)

t (hours) = 0.5

A longer window means a bigger mean and a wider spread.

Worked example2 calls per hour

P(none in 30 min)? P(more than 8 in 3 h)?

Goal: Two probabilities.
Strategy: Scale $\lambda$ .

$\operatorname{Po}(1):\ P(0)=e^{-1}=0.368$
Half an hour.
$\operatorname{Po}(6):\ P(X>8)=1-P(X\le8)=0.153$
Three hours.

Answer

0.368

;

0.153

✓ CheckLonger window, bigger mean ✓.

Worked exampleFinding the rate

P(no faults in 1 m of cable) $=0.2$ . Faults per metre?

Goal: $\lambda$ .
Strategy: $P(0)=e^{-\lambda}$ .

$e^{-\lambda}=0.2$
Condition.
$\lambda=\ln5=1.61$
Take logs.

Answer

1.61

per metre.

✓ Check

e^{-1.61}=0.2

✓.

Worked exampleSeveral separate intervals

Rate 2 per hour. P(exactly 1 call in each of 3 separate hours)?

Goal: Probability.
Strategy: Independent: multiply.

$2e^{-2}=0.2707$
One hour.
$0.2707^3=0.0198$
Three hours.

Answer

0.0198

✓ CheckDifferent from "exactly 3 in 3 hours" (

0.0892

), which allows uneven splits ✓.

Pause and think

Rate 4 per day. Distribution for a week?

Summary

$\lambda$ scales with interval.
Independent intervals multiply.

Rate: Mean per unit.

HistoryCounting the stars' flashes. In 1910, Ernest Rutherford and Hans Geiger counted alpha particles hitting a screen in

7.5

-second intervals,

2608

times. Their counts fitted a Poisson distribution almost perfectly. Radioactive decay is the textbook Poisson process: atoms decay independently, at random, at a constant rate.

Fun factClusters happen by chance. Random events cluster more than people expect. During the Blitz in 1944, Londoners believed V-1 flying bombs were targeting certain districts. The statistician R. D. Clarke divided south London into

576

squares and showed the hits per square followed a Poisson distribution: the clusters were exactly what pure randomness produces.

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