6.1 The Poisson Distribution

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1. What Is the Poisson Distribution?

The Poisson distribution (say: "pwah-SON") is a special type of probability distribution. It tells you the probability of a certain number of events happening in a fixed period of time or space, when those events happen randomly and independently of each other.

For example:

• The number of cars passing a junction per minute
• The number of phone calls received by an office per hour
• The number of typing errors per page of text
• The number of radioactive particles detected per second

We write: X ~ Po(λ) — this means "X follows a Poisson distribution with parameter λ (lambda)."

λ (lambda) is the mean — the average number of events you expect in the given interval.

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2. When Can You Use the Poisson Distribution as a Model?

Before using the Poisson distribution, you need to check that the situation fits. The Poisson distribution is appropriate when:

• Events happen randomly — you cannot predict exactly when or where each event will occur
• Events happen independently — one event happening does not make another more or less likely
• Events happen at a constant average rate — the mean number of events per interval stays roughly the same
• Two events cannot happen at exactly the same instant (or in the same tiny point of space)

Practical check: If you have data, calculate the mean and the variance of the data. For a Poisson distribution, these should be roughly equal. If the mean ≈ variance, a Poisson model is likely appropriate.