Poisson distribution

A family of theoretical distributions that is useful as a model for some discrete random variables. Each distribution in this family gives the probability of obtaining a specified number of occurrences of a phenomenon in a specified interval in time or space, under the following conditions:

• On average, the phenomenon occurs at a constant rate, λ.
• Occurrences of the phenomenon are independent of each other.
• Two occurrences of the phenomenon cannot happen at exactly the same time or in exactly the same place.

Each member of this family of distributions is uniquely identified by specifying λ. As such, λ, is the parameter of the Poisson distribution and the distribution is sometimes written as Poisson(λ).

Let random variable X represent the number of occurrences of a phenomenon that satisfies the conditions stated above. The probability of x occurrences is calculated by:

P(X = x) = 

for x = 0, 1, 2, ...

Example

A graph of the probability function for the Poisson distribution with λ = 3 is shown below.

Curriculum achievement objectives reference

Probability: Level 8

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