The Poisson distribution arises when counting events that occur independently and randomly at a constant rate over time or space. The number of events occuring during a fixed period or in a fixed region is a Poisson(λ) random variable where λ is the expected number of events. Because Poisson random variables count the number of events occuring, they can only take integer values in the range 0, 1, 2, ...
| Parameter | Range | Description |
|---|---|---|
| λ | λ ≥ 0 | Rate parameter |
Probability Mass Function
Support
Mean
Variance
| Example | λ |
|---|---|
| On average, a lottery jackpot is won every four weeks. Let X be the number of jackpot winners in a given week. | 0.2500 |
| On average, an accident occurs at a traffic intersection every six months. Let X be the number of accidents in one year. | 2.000 |
| On average, 28 people are killed by lightning in the US each year. Let X be the number of people killed in a year. | 28.00 |
X ~ Poisson(λ)
E(X) = , Var(X) =
A particular feature of the Poisson distribution is that the variance equals the mean. Sample counts therefore vary more as the mean increases.
The illustration above shows events occuring randomly in time over a fixed period independently of other events. The expected number of events over this period is given by λ. The actual number of events occurring is a random variable X, which has a Poisson(λ) distribution.
The simulation above shows events occuring randomly in time over a fixed period independently of other events, where λ is the expected number of events over the whole period. The random variable X is the actual number of events occurring during this period, which has a Poisson(λ) distribution. The histogram accumulates the results of each simulation.