Poisson & Binomial Distributions

Quick Summary

• The Poisson distribution, Po(λ), models the number of random events happening in a fixed interval of time or space.
• Three conditions must hold: events happen independently, singly (never simultaneously), and at a constant average rate.
• For X ~ Po(λ): mean = variance = λ. This is the Poisson's signature feature.
• Scaling the interval scales λ by the same factor (e.g. double the time → double λ).
• Independent Poisson variables add together: if X ~ Po(λ) and Y ~ Po(μ), then X + Y ~ Po(λ + μ) — but only over the same interval.
• A Binomial B(n, p) can be approximated by Po(np) when n is large (n > 30) and p is small (np ≤ 10).

What Is the Poisson Distribution?

Imagine you're standing by a busy road counting how many buses pass in an hour, or counting how many typos are scattered across a page of text. In both cases you're not asking "will it happen or not" (that's a yes/no world — Binomial territory). You're asking "how many times will this happen?" when the event could, in theory, happen any number of times — 0, 1, 2, 3... all the way up.

That's exactly what the Poisson distribution is built for: counting random occurrences within a fixed window — a window of time (calls to a call centre per hour) or a window of space (flowers per square metre of a field).

The Three Conditions

For a Poisson model to be valid, all three of these need to be roughly true:

• Independence — one occurrence doesn't make another more or less likely. One customer walking into a shop shouldn't affect whether the next one does.
• Singly — events can't happen at exactly the same instant (in time) or exactly the same point (in space). Two things can't occur "simultaneously."
• Constant average rate — the rate of events doesn't change across the interval. Rush hour vs. 3am traffic would violate this if your interval spans both.

When the Conditions Break Down

Examiners love asking you to criticise a Poisson model — meaning, explain why the real-world situation might not actually satisfy those three conditions. Classic example: counting people entering a restaurant.

• Not independent — people often arrive because a friend invited them, so arrivals are linked.
• Not singly — families or groups walk in together, arriving at the "same" moment.
• Not constant rate — dinner time sees way more arrivals than 3pm on a quiet afternoon.