Introduction to the Poisson Distribution
The Poisson distribution is one of the most important discrete probability distributions in statistics, especially for modelling rare or random events.
Poisson Distribution: A discrete probability distribution that gives the probability of a given number of events occurring in a fixed interval of time or space, where events occur at a known constant mean rate and independently of one another.
Typical real-world examples include:
- Number of emails arriving in an inbox per hour
- Number of road accidents at a junction per week
- Number of radioactive decays per second
- Number of misprints per page in a book
The distribution is named after French mathematician Siméon Denis Poisson, who introduced it in the 19th century. It is particularly well-suited to modelling rare events , situations where an event could happen many times, but typically occurs infrequently.
Think of the Poisson distribution like counting the number of buses that arrive at a stop in a 10-minute window. You know roughly how many to expect on average, but the exact count varies randomly from window to window , and crucially, one bus arriving doesn't cause or prevent another.
Conditions for Using the Poisson Distribution
Before applying the Poisson distribution, you must verify that the situation satisfies two key conditions:
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Independence of Events: The occurrence of one event must have no effect on the probability of any other event occurring. Events happen independently of each other.
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Constant Average Rate: Events must occur at a uniform average rate throughout the interval. The rate does not speed up or slow down within the period being considered.
A third implied condition is that non-overlapping intervals are independent , what happens in one time interval does not affect what happens in a separate, non-overlapping interval.
Checking conditions: A coffee shop records that, on average, 20 customers arrive between 2 PM and 3 PM on weekdays. Each customer's decision to visit is independent, and arrivals occur at a roughly constant rate throughout that hour.
✔ Independence: customers arrive independently.
✔ Constant rate: no systematic rush within the hour.
This can be modelled as .
If the rate is not constant (e.g. customer arrivals spike sharply at lunchtime), the Poisson model is not appropriate for the full period. You would need to restrict the model to a sub-interval where the rate is approximately constant.