Poisson Thinning

Randomly keep each Poisson event with probability $p$ : you get a Poisson process of rate $λ p$ .

What is it?

If events arrive as Poisson( $λ$ ) and each is independently of type A with probability $p$ , type A and type B streams are independent Poisson processes with rates $λ p$ and $λ (1 - p)$ .

Why is it useful?

Splitting arrivals into buyers/sellers or success/failure streams is common in trading and probability interviews.

When should you think of it?

Use it when Poisson arrivals are independently classified into types.

Think of this method when…

Poisson arrivals with a rate
Each arrival independently labeled or filtered
Questions about one type only
Merging independent streams (superposition, the reverse)

Core idea

Thinned streams are Poisson and, surprisingly, independent of each other.

N_{A} \sim Poisson (λ pt), N_{B} \sim Poisson (λ (1 - p) t), N_{A} ⊥ N_{B}

Step-by-step playbook

1Identify the base rate $λ$ and the keep probability $p$ .
2Write the thinned rate $λ p$ .
3Answer using the Poisson distribution with that rate.

Worked example

Customers arrive at 10 per hour; each buys with probability $0.3$ independently. What is the probability that no one buys in an hour?

1
Buyers form a Poisson process of rate $10 \times 0.3 = 3$ per hour.
2
Probability of zero events.
$P (N = 0) = e^{- 3} \approx 0.0498$
Answer: $e^{- 3} \approx 0.050$

Common mistakes

Assuming the two thinned streams are dependent.
Thinning with probabilities that depend on other arrivals.

Would you use this method?

A fixed number, 20, of customers arrive, each buying with probability 0.3. Is the number of buyers Poisson?

Practice this method

14 QuantSolve questions are tagged with this method.

Practice this method

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Thinning & Superposition · Stochastic Processes Poisson · Distributions Poisson Processes · Stochastic Processes

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