Math Atom · M-C05

The Poisson distribution

8 minMath AtomProbability

Know these first

By the end you will be able to

  1. 1Recognise a Poisson setting: events arriving at random, independently, at a steady average rate, counted over a fixed interval.
  2. 2Compute P(N = k) = λᵏe^(−λ)/k!, cumulative probabilities, and the chance of at least one event, 1 − e^(−λ).
  3. 3Use mean = variance = λ, and scale λ with the length of the interval.
  4. 4See the Poisson as the limit of a binomial with many trials and a small probability, and check counts for overdispersion.

Why you need it

RiskAtoms Bank has booked 12 operational losses of £1m or more in 10 years. What is the chance of none next year? Of three or more? One number, the average rate, answers both, if the losses arrive like a Poisson count.

These Core Path lessons use it:

  • L3.7.07: Measuring frequency and severity

    Models how many losses of each event type happen in a year as a Poisson count, fitted as the average count, before adding up their sizes.

  • L3.7.06: Scenario analysis

    Turns a workshop's “once in N years” into a rate λ = 1/N, and the chance of at least one such event in a year.

  • L3.7.08: Cyber risk

    Cyber losses as how often attacks succeed times how much each costs: the frequency is a Poisson rate.

You need powers from M-A03: Exponents, the number e from M-A05: The number e and continuous growth, expectation and variance from M-C02: Expectation and M-C03: Variance and standard deviation, and the binomial count from M-C04: The binomial distribution. The size of each loss is a separate question, often answered with M-C07: The lognormal distribution.

The idea

Events at random moments

A binomial count needs a fixed number of trials. Many risks have none: a system can fail at any moment of the year, not on day 17 or not. What stays fixed is the length of the interval and the average rate. Take the book's system failures, 5.1 a year (illustrative), and chop the year into slots. Count a slot as a success if anything happened in it, and you are back to a binomial with p = λ/n.

Chop the year finer

business disruption and system failures: 5.1 a year

Slots in the year

JFMAMJJASOND
Events in year 14
p = λ / n for each slot0.425
Slots with an event (the binomial count)4
Slots holding two or more0

Dots are events at random moments. A slot counts once however many events land in it, so shaded slots (two or more) make the binomial count too low. Finer slots make that rarer.

Poisson probability of k events binomial, 12 slots
01234567891011121314

Events in a year, k. Largest gap between a dot and its bar: 0.0529.

With months as slots, two failures often share a month, and the binomial dots sit visibly off the bars. With days or hours that almost never happens: the largest gap shrinks from 0.053 with 12 slots to 0.00005 with 8,760. The bars they settle on are the Poisson distribution: the count of events in an interval, fixed by one number, the rate λ.

The rules

The setting. Each occurrence is an event. Four conditions make their count Poisson.

The probabilities. A power, a factorial and e. Adding them up gives the cumulative probability.

Where it comes from. The idea above, in symbols: a binomial with ever more slots and an ever smaller probability.

Mean, variance and time. One number is both the mean and the variance, and it scales with the interval. That gives a quick check on data, the dispersion ratio, and a name for its failure, overdispersion.

The rules, for reference

N∼Poisson(λ):P(N=k)=λke−λk!,k=0,1,2,…N \sim \text{Poisson}(\lambda): \quad P(N = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots
E[N]=Var⁡(N)=λP(N≥1)=1−e−λλt=rate×tE[N] = \operatorname{Var}(N) = \lambda \qquad P(N \ge 1) = 1 - e^{-\lambda} \qquad \lambda_t = \text{rate} \times t
Bin(n,p)≈Poisson(np)for large n and small p\text{Bin}(n, p) \approx \text{Poisson}(np) \quad \text{for large } n \text{ and small } p

In the calculator: poisspmf(k, λ) gives P(N = k), poisscdf(k, λ) gives P(N ≤ k), and exp and factorial (or k!) build the formula by hand.

Try it

Choose a yearly rate, from the book's presets or the slider, and an interval. Things to try: switch from a year to a month and watch the bars slide left and bunch up; find the chance of at least one large loss in a quarter; simulate 10 years, then 1,000, and watch the sample variance settle on the mean.

Try itPoisson explorer

Open the distribution calculator →

The book's yearly rates (illustrative)

Interval

Event

P(N = k), λ = 1.2 share of 10 simulated yearsdashed line: mean; shading: mean ± 1 sd
012345678910

Events per year, k from 0 to 10; bright bars are in the event. Outside this range the probabilities are negligible.

Simulated years10
Sample mean1.5
Sample variance (n − 1)1.39
First counts2, 4, 0, 1, 0
Formula: mean = variance1.2
Variance / mean0.93
λ=1.2=1.2 per year\lambda = 1.2 = 1.2 \text{ per year}
P(N≥1)=1−P(N≤0)=0.6988P(N \ge 1) = 1 - P(N \le 0) = 0.6988

Probability

0.6988

As a percentage69.88%
Mean λ1.2
Variance λ1.2
sd √λ1.095
P(at least one in a year)69.88%

Worked examples

1. Large losses next year

Step 1 of 6
  1. 1

    The set-up

    In the Module 3.7 book (illustrative), RiskAtoms Bank booked 12 net losses of £1m or more in 2016–2025. Treat next year's count N as Poisson with λ = 12 / 10 = 1.2.

2. External fraud: a year and a month

Step 1 of 4
  1. 1

    The set-up

    The book's most frequent type is external fraud: 35.3 losses a year on average over 10 years. Its busiest year, 2021, had 49. Was that extraordinary? And how does the count look month by month?

3. Is a Poisson good enough?

Step 1 of 6
  1. 1

    The set-up

    A Poisson's variance equals its mean. Do the book's yearly counts of external fraud losses behave like that?

    Year2016201720182019202020212022202320242025
    Losses36322926334938313544

Data: the Module 3.7 operational risk book, RiskAtoms Bank plc, 2016–2025, losses with a net loss above zero by accounting year. Illustrative, not real loss history.

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 2

A desk has on average 2 trade booking errors a month, arriving as a Poisson count. What is the probability of a month with none? Give a decimal.

Question 2 · objective 2

N ~ Poisson(3). What is P(N = 2)? Give a decimal.

Question 3 · objective 3

Losses of a certain type arrive at 24 a year on average, as a Poisson count. What is the standard deviation of the number in one quarter?

Question 4 · objective 4

A payments system processes 1,000 payments an hour, each independently failing with probability 0.002. Which distribution is a good approximation for the number of failures in an hour?

Question 5 · objective 4

Twelve years of yearly counts for one loss type have mean 10 and sample variance 30. What does this suggest?

Glossary

Event
One occurrence of the thing being counted, such as one operational loss of £10,000 or more.
Poisson distribution
The distribution of the number of events N in a fixed interval when events arrive at random, independently, at a steady average rate λ. Written Poisson(λ).
Rate λ
λ (lambda): the average number of events per interval. It is the Poisson's mean and also its variance.
Factorial
k! = k × (k − 1) × … × 2 × 1, with 0! = 1. For example 4! = 24.
Dispersion ratio
Sample variance divided by sample mean of a set of counts. About 1 for Poisson counts, give or take sampling noise.
Overdispersion
Counts that vary more than a Poisson allows (variance well above the mean), a sign that events cluster or the rate itself changes.

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