Core Process

6-10 Hours

Poisson Processes

The fundamental counting process for modeling rare events in continuous time

Poisson Process Calculator

Calculate probabilities for Poisson process increments

Calculation Type

Rate λ

Time Interval (t)

Number of Events (k)

Definition & Axioms - Axiomatic Definition

A counting process $\{N(t), t \geq 0\}$ is a Poisson process with rate $\lambda > 0$ if:

$N(0) = 0$
Independent increments
For $s < t$ , $N(t) - N(s) \sim \text{Poisson}(\lambda(t-s))$

Key Properties - Inter-Arrival Times

The inter-arrival times $T_n$ are i.i.d. exponential( $\lambda$ ): $T_n \sim \text{Exp}(\lambda)$ .

Mean & Variance

$E[N(t)] = \lambda t$ and $\text{Var}(N(t)) = \lambda t$ .

Fundamental Theorems - Superposition Theorem

The superposition of $k$ independent Poisson processes with rates $\lambda_1, \lambda_2, \dots, \lambda_k$ is a Poisson process with rate $\lambda = \sum_{i=1}^k \lambda_i$ .

Thinning (Decomposition) Theorem

If each event in a Poisson( $\lambda$ ) process is independently classified into category $i$ with probability $p_i$ , then each resulting sub-process is an independent Poisson process with rate $\lambda p_i$ .

Worked Examples - Example 1: Probability of No Events

Problem:

For a Poisson process with rate $\lambda = 2$ events/hour, find $P(N(1) = 0)$ .

Solution:

$P(N(1) = 0) = e^{-\lambda t} = e^{-2 \times 1} = e^{-2} \approx 0.1353$ .

Example 2: Superposition

Problem:

Two independent Poisson processes have rates $\lambda_1 = 3$ and $\lambda_2 = 5$ . What is the rate of their superposition?

Solution:

The superposition is a Poisson process with rate $\lambda = \lambda_1 + \lambda_2 = 3 + 5 = 8$ .

Example 3: Thinning

Problem:

Customers arrive at rate $\lambda = 10$ per hour. 30% buy coffee. What is the rate of the coffee purchase process?

Solution:

By thinning, the coffee purchase process is Poisson with rate $\lambda p = 10 \times 0.3 = 3$ per hour.

Practice Quiz

Questions

Correct

Accuracy

What is the probability mass function for a Poisson process increment

N(t) - N(s)

Not attempted

What is the distribution of inter-arrival times in a Poisson process?

Not attempted

What is

P(N(t) = 0)

for a Poisson process with rate

\lambda

Not attempted

What is the superposition of two independent Poisson processes with rates

\lambda_1

and

\lambda_2

Not attempted

What is the mean of

N(t)

for a Poisson process with rate

\lambda

Not attempted

What is the variance of

N(t)

for a Poisson process with rate

\lambda

Not attempted

What is the distribution of the time to the

n

-th event

W_n

in a Poisson process?

Not attempted

What is 'thinning' in the context of Poisson processes?

Not attempted

What is the difference between a homogeneous and non-homogeneous Poisson process?

Not attempted

Given

N(t) = n

, what is the distribution of the event times?

Not attempted

Frequently Asked Questions

Why are inter-arrival times exponentially distributed?

The exponential distribution is the only continuous memoryless distribution. Since the Poisson process has independent increments (the memoryless property), and events occur at a constant rate λ, the time until the next event must follow an exponential distribution with parameter λ. This is a direct consequence of the process's fundamental properties.

What's the difference between a homogeneous and non-homogeneous Poisson process?

A homogeneous Poisson process has a constant rate λ(t) = λ, meaning events occur at the same average rate over time. A non-homogeneous (or inhomogeneous) process has a time-varying rate λ(t), allowing the intensity to change—like modeling rush hour traffic where the arrival rate increases during peak hours.

How does the superposition theorem work?

If you have k independent Poisson processes with rates λ₁, λ₂, ..., λₖ, their superposition (sum) is also a Poisson process with rate λ₁ + λ₂ + ... + λₖ. This is powerful for modeling: if two independent service centers generate customers at rates 3/hr and 5/hr, the combined arrival stream is Poisson with rate 8/hr.

What is meant by 'thinning' a Poisson process?

Thinning (or decomposition) is when you independently classify each event in a Poisson(λ) process into categories with probabilities p₁, p₂, ..., pₖ. Each resulting sub-process is an independent Poisson process with rate λpᵢ. For example, if customers arrive at rate 10/hr and 30% buy coffee, the coffee purchase process is Poisson(3).

Why is P(N(t)=0) = e^{-λt} important?

This gives the probability of no events in time [0,t]. It's fundamental for reliability theory (probability a component survives time t without failure) and queuing theory (probability a server remains idle). It's also the starting point for deriving the full Poisson distribution recursively.