Random events on a timeline

Emails, phone calls, and earthquakes all arrive at unpredictable times. The Poisson process is the mathematical model for events scattered along a timeline.

What do you think?

You receive emails at an average rate of 3 per hour. What's the probability of receiving exactly 0 emails in the next 20 minutes?

What is a Poisson process?

Poisson Process

A Poisson process with rate $\lambda > 0$ is a counting process $N(t)$ where:

$N(0) = 0$ (start at zero)
Independent increments: counts in non-overlapping intervals are independent
Stationary increments: $N(t+s) - N(t) \sim \text{Pois}(\lambda s)$ for any $t$
Events happen one at a time (no simultaneous arrivals)

The number of events in a time interval of length $s$ follows a Poisson distribution with parameter $\lambda s$ .

We already met the Poisson distribution in the discrete chapter. Now we see where it naturally arises — as the counting distribution for a continuous-time process.

Watch events arrive

Click to start and watch random events appear on a timeline. The count follows a Poisson distribution:

Poisson Process Timeline

Rate λ2.0

N(10)

Expected λt

20.0

Avg inter-arrival

—

Expected 1/λ

0.500

Inter-arrival times

Inter-arrival Times

The time between consecutive events in a Poisson process with rate $\lambda$ follows an Exponential( $\lambda$ ) distribution: $T_i \sim \text{Expo}(\lambda), \quad E[T_i] = 1/\lambda$

This connects three things we've already learned: the Poisson distribution gives the number of events in a fixed time, the Exponential distribution gives the time between events, and the memoryless property means the process restarts after each event.

Why Exponential inter-arrivals?

P(T_1 > t) = P(N(t) = 0) = e^{-\lambda t}

The first arrival time T₁ satisfies

Step 1 of 4

Simulate Poisson process events and measure the gaps between them — watch the histogram match the Exponential PDF:

Inter-Arrival Times

Measure the gaps between consecutive Poisson process events — they follow Exponential(λ)

Rate λ = 3.0: 3.0

0.58

Gap histogram Exp(λ) PDF

Gaps measured

Sample mean

0.0000

E[gap] = 1/λ

0.3333

Properties

Property	Formula
Count in $[0, t]$	$N(t) \sim \text{Pois}(\lambda t)$
Expected count	$E[N(t)] = \lambda t$
Inter-arrival time	$T_i \sim \text{Expo}(\lambda)$
$n$ -th arrival time	$S_n \sim \text{Gamma}(n, \lambda)$
Merging	Rate $\lambda_1$ + Rate $\lambda_2$ = Rate $\lambda_1 + \lambda_2$
Splitting with prob $p$	Rate $\lambda$ → Rate $p\lambda$

Practice problems

Customers arrive at rate λ = 5/hour. Expected customers in 2 hours? (whole number)

Same arrival rate. What's the expected time between customers (in minutes)? (whole number)

If emails arrive at rate 3/hour and texts arrive at rate 2/hour (independently), what's the combined rate of notifications? (whole number)

Summary

Concept	Key Idea
Poisson process	Counting random events with rate $\lambda$
Count distribution	$N(t) \sim \text{Pois}(\lambda t)$
Inter-arrivals	$T_i \sim \text{Expo}(\lambda)$ , independent
Memoryless	Waiting time doesn't depend on how long you've waited

The Poisson process is the continuous-time analog of the Bernoulli process. Bernoulli = coin flips at discrete times, Poisson = events at continuous times.

What's next

We'll see what happens when Poisson processes combine or split in superposition and thinning.