Poisson

The distribution of counts of rare, independent events occurring at a constant average rate.

1. The idea

A Poisson $(λ)$ random variable counts the number of events in a fixed interval when events occur independently at a constant average rate $λ$ . Its PMF is $P (X = k) = e^{- λ} λ^{k} / k!$ for $k = 0, 1, 2, \dots$

It arises as the limit of Binomial $(n, p)$ as $n \to \infty$ , $p \to 0$ with $n p \to λ$ fixed — many independent trials, each individually rare, combine into a Poisson count.

A hallmark identity is that mean equals variance: $E [X] = Var (X) = λ$ . Sums of independent Poissons are Poisson with summed rates, which connects directly to Poisson processes.

2. Why it matters

Order arrivals, trade counts, and rare-event questions are almost always Poisson in disguise; recognizing the rate parameter $λ$ (often "events per unit time $\times$ time") lets you skip a full derivation.

3. Core mathematics

Parameter

λ > 0

mean rate over the interval

Support

k \in {0, 1, 2, \dots}

PMF

P (X = k) = \frac{e ^{- λ} λ ^{k}}{k !}

Expectation

E [X] = λ

Variance

Var (X) = λ

Key property

X_{1} \sim Poisson (λ_{1}), X_{2} \sim Poisson (λ_{2}) independent \Rightarrow X_{1} + X_{2} \sim Poisson (λ_{1} + λ_{2})

4. How to recognize it

Think of this when…

"Number of events/arrivals/defects in a given time or space interval"
A rate is given per unit time (or area), and events are independent
Binomial with very large $n$ and very small $p$ — a rare-event approximation
Questions combining or splitting rates across independent counting processes

5. Worked example

Worked example

Orders arrive at a desk at an average rate of $4$ per minute, Poisson distributed. Find the probability that exactly $2$ orders arrive in a 30-second window.

1
Scale the rate to the window length: $30$ seconds is half a minute, so $λ = 4 \times 0.5 = 2$ .
2
Let $X$ be the count in that window; $X \sim Poisson (2)$ .
3
Apply the PMF.
$P (X = 2) = \frac{e ^{- 2} 2 ^{2}}{2 !} = \frac{4 e ^{- 2}}{2} = 2 e^{- 2}$
4
Evaluate numerically.
$2 e^{- 2} \approx 0.2707$
Answer: $P (X = 2) \approx 0.271$ .

6. Common mistakes

Forgetting to rescale $λ$ to match the time window given in the question
Treating a Poisson count as if it had a fixed number of trials $n$
Adding independent Poisson rates when the events are not actually independent
Confusing $λ$ (a rate, possibly per unit time) with $E [X]$ over a different interval

7. Quick check

If calls arrive at rate $6$ per hour, what is the expected number of calls in a 10-minute period?

For $X \sim Poisson (3)$ , which statement is true?

8. Practice

2 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Poisson Thinning

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