Poisson distribution
High school
Consider events that happen independently of each other at an average rate of $\lambda$ per unit of time, such as customers arriving at a shop. The number $X$ of events in one unit of time has a Poisson distribution with mean $\lambda$:
\[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad \text{for } k = 0, 1, 2, \dots.\]Examples
Worked examples
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Customers arriving at a shop independently of each other at an average rate of $3$ per hour
: The number of customers in an hour: Poisson with mean $3$
Undergraduate
For customers arriving independently at an average rate of $3$ per hour, the number of customers in an hour has a Poisson distribution with mean $3$, because the number counts independent events at a constant rate.
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Emails arriving independently of each other at an average rate of $6$ per hour, and the event that no email arrives in the next $10$ minutes
: The number $N$ of emails in $10$ minutes is Poisson with mean $1$, so $P(N = 0) = e^{-1} \approx 0.368$
Undergraduate
For emails arriving at $6$ per hour, the number of emails in $10$ minutes is a count in a fixed time, so the number has a Poisson distribution with mean $6 \cdot \frac{10}{60} = 1$.
Properties
- A Poisson random variable with mean $\lambda$ also has variance $\lambda$.
- The Poisson distribution is the limit of the binomial distribution with parameters $n$ and $\lambda / n$ as $n \to \infty$.
Distinctions involving this term
- Poisson vs. exponential distribution: Consider events that happen independently of each other at a constant average rate $\lambda$, such as customers arriving at a shop (a Poisson process). The Poisson distribution describes the number of events in a fixed length of time, which is a whole number. The exponential distribution describes the time until the next event, which is a positive real number.