The Poisson Distribution

The Poisson distribution is one of the most important distributions in Further Statistics. It models the number of events occurring in a fixed interval of time or space when the events happen independently at a constant average rate. This lesson covers the derivation, properties, and applications of the Poisson distribution.

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When to Use the Poisson Distribution

The Poisson distribution is appropriate when:

Condition	Meaning
Events occur singly	Two events cannot happen at exactly the same instant
Events occur independently	One event does not affect the probability of another
Events occur at a constant average rate	The mean rate $\lambda$λ does not change over the interval
The probability of an event in a small interval is proportional to the length of the interval	$P(\text{event in } \delta t) \approx \lambda \delta t$P(event in δt)≈λδt

Examples of Poisson-distributed variables:

• Number of emails received per hour
• Number of typing errors per page
• Number of cars passing a point per minute
• Number of radioactive decays per second

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The Probability Formula

If $X \sim \text{Po}(\lambda)$X∼Po(λ), then: