Poisson Process — Definition, Formula & Examples

A Poisson process is a model for counting random events that occur independently at a constant average rate over time (or space). Each count in a fixed interval follows a Poisson distribution, and the waiting time between consecutive events follows an exponential distribution.

A Poisson process with rate $\lambda > 0$ is a counting process $\{N(t) : t \geq 0\}$ satisfying: (1) $N(0) = 0$ ; (2) it has independent increments; (3) for any interval of length $t$ , the number of events $N(t)$ follows a Poisson distribution with parameter $\lambda t$ , so $P(N(t) = k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}$ for $k = 0, 1, 2, \ldots$

Key Formula

P(N(t) = k) = \frac{(\lambda t)^k \, e^{-\lambda t}}{k!}

Where:

$N(t)$ = Number of events occurring in an interval of length t
$\lambda$ = Average rate of events per unit time
$t$ = Length of the time interval
$k$ = Specific number of events (non-negative integer)

How It Works

You use a Poisson process whenever events happen randomly and independently at a steady average rate. The rate parameter

\lambda

tells you how many events to expect per unit of time. To find the probability of exactly

k

events in an interval of length

t

, plug into the Poisson formula with mean

\lambda t

. The gaps between successive events are independent and exponentially distributed with mean

1/\lambda

. If the rate changes over time, you need a non-homogeneous Poisson process instead.

Worked Example

Problem: A call center receives calls at an average rate of 3 per minute. What is the probability that exactly 5 calls arrive in a 2-minute window?

Identify parameters: The rate is λ = 3 calls/min and the interval is t = 2 min, so the expected number of calls is λt = 6.

\lambda t = 3 \times 2 = 6

Apply the formula: Substitute λt = 6 and k = 5 into the Poisson probability formula.

P(N(2) = 5) = \frac{6^5 \, e^{-6}}{5!}

Compute: Calculate each piece: 6⁵ = 7776, 5! = 120, and e⁻⁶ ≈ 0.002479.

P = \frac{7776 \times 0.002479}{120} \approx \frac{19.277}{120} \approx 0.1606

Answer: The probability of exactly 5 calls in 2 minutes is approximately 0.161, or about 16.1%.

Visualization

Why It Matters

Poisson processes appear throughout queueing theory, reliability engineering, and network traffic modeling. Insurance companies use them to model claim arrivals, and biologists use them to describe mutations along a genome. Mastering this concept is essential in any probability or stochastic processes course.

Common Mistakes

Mistake: Forgetting to multiply λ by t when the interval is not one unit long.

Correction: The Poisson parameter for an interval of length t is λt, not λ. Always scale the rate by the interval length before plugging into the formula.