Merging streams

Two lanes of traffic merge onto a highway. If each lane is a Poisson process, what's the result? Another Poisson process — with the rates added.

What do you think?

Cars arrive on Lane A at rate 10/min and Lane B at rate 6/min (both Poisson). After merging, the combined stream has rate ___/min.

whole number

Superposition (merging)

Superposition

If $N_1(t)$ and $N_2(t)$ are independent Poisson processes with rates $\lambda_1$ and $\lambda_2$ , then: $N(t) = N_1(t) + N_2(t) \sim \text{Poisson process with rate } \lambda_1 + \lambda_2$

This generalizes: merging $k$ independent Poisson streams with rates $\lambda_1, \ldots, \lambda_k$ gives rate $\sum \lambda_i$ .

Watch two streams of events merge into one:

Poisson Superposition & Thinning

λ_A2

λ_B3

Thinning (splitting)

Thinning

If $N(t)$ is a Poisson process with rate $\lambda$ , and each event is independently "kept" with probability $p$ (or "discarded" with probability $1-p$ ), then:

Kept events: Poisson process with rate $p\lambda$
Discarded events: Poisson process with rate $(1-p)\lambda$
The two are independent.

Why thinning works

P(\text{event in } dt) = \lambda \, dt

In a small interval dt, an event occurs with probability λdt

Step 1 of 4

Thinning is the reverse of superposition. Superposition combines independent streams; thinning splits one stream into independent sub-streams.

See thinning in action — each event is independently kept or discarded, creating two independent Poisson sub-streams:

Poisson Thinning

Each event is independently kept (green) with probability p or discarded (red) with probability 1−p

Original rate λ = 8: 8

220

Keep probability p = 0.40: 0.40

0.10.9

Trials

Avg kept

0.00

λp (theory)

3.2

Avg discarded

0.00

λ(1−p) (theory)

4.8

Real-world examples

Scenario	Process	Operation
Email + Slack notifications	Two Poisson streams	Superposition → combined alert rate
Customers: 30% use coupon	One Poisson stream	Thinning → coupon users are Poisson
Earthquakes across regions	Regional processes	Superposition → global earthquakes
Quality control: 5% defect rate	Production line	Thinning → defects are Poisson
911 calls: fire vs. medical	Combined calls	Thinning → each type is Poisson

Conditional arrival location

Uniformity Property

Given that exactly $n$ events occur in $[0, T]$ , their arrival times are distributed as $n$ independent Uniform $(0, T)$ random variables (sorted).

Conditioning on the count, the arrival locations are uniformly distributed with no clustering or gaps.

Practice problems

Three independent Poisson processes have rates 2, 3, and 7. The merged process has rate ___. (whole number)

Emails arrive at rate 10/hour. You read 40% of them. The process of read emails has rate ___/hour. (whole number)

After thinning a Poisson process, are the kept and discarded streams independent? (yes or no)

Summary

Concept	Key Formula
Superposition	$\lambda_1 + \lambda_2$ Poisson processes merge to rate $\lambda_1 + \lambda_2$
Thinning	Keep with prob $p$ → rate $p\lambda$
Independence	Kept and discarded streams are independent
Uniformity	Given $n$ events in $[0,T]$ , locations are Uniform $(0,T)$

Superposition and thinning make Poisson processes composable. You can build complex event models by merging and splitting simple Poisson streams, and the result is always Poisson.

Course complete

You've covered counting, distributions, limit theorems, Markov chains, MCMC, and Poisson processes — enough to read most applied probability and statistics.