poisson distribution

statistics

Poisson distributions are used when there are a number of rare events that occur during a specific period/area, like “number of traffic accidents per month at a busy intersection” or “number of unscheduled admissions per day to a hospital”.

Characteristics of a Poisson Random Variable

1. The experiments consists of counting the number of times a certain event occurs during a given unit of time, area, or volume.
2. The probabiliy that an event occurs during a unit of time/area/volume is the same for all units.
3. The number of events that occur aer independent of the number that occur in other units.
4. The mean (or expected) number of events in each unit is denoted by lambda ($\lambda$).

Probability distribution, mean, and variance for Poisson random variables.

\begin{math} p(x) = \frac{\lambda^x e^{-\lambda}}{x!} (x = 0,1,2,…)\\ \mu = \lambda\\ \sigma^2=\lambda \end{math}

where $\lambda$ is the mean number of events during a given unit of time/area/volume.