Poisson Distribution

The Poisson Distribution is a Probability Mass Function that models the Probability of a given number of events occurring in a fixed interval of time such as:

Example

• Number of orders in an hour at a restaurant
• Heart rate in beats per minute
• Goals scored per football season

The poisson distribution models a Bernoulli Distribution where $n$ is very large and $p$ is close to either $0$ or $1$.

The time between two consecutive events in a poisson process follows an exponential distribution.

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Formula:

$Pois(x)=\frac{e^{-\lambda }{\lambda }^k}{k!}$

Where

• $\lambda$ is the average rate of occurrence of an event
  • $\lambda$ may be given by $np$ (see binomial distribution)
• $k$ is the number of events

Note

The expected value and variance of a poisson distribution are both $\lambda$.

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Poisson Process

The formula for exactly $k$ events occurring in a given time $T$ (poisson process) is given by: $P(X=k)=\frac{(\lambda t{)}^k{e}^{-\lambda t}}{k!}$

Where

• $\lambda$ is the average rate of occurrence of an event
• $k$ is the number of events
• $t$ is the amount of time periods experienced