Statistics for Maintenance Modelling: Continuous Lifetime Distributions 1

# The Poisson Distribution¤

The Poisson distribution is often used to model how many failures occur within a time interval where it is assumed that:

- the number of failures is proportional to the length of the interval
- the number of failures in one time interval does not affect the number of failures in another interval.

A typical situation would be a complex system that is made up of many similar but not necessarily identical components. Although each component is reliable, their large number means that there is a chance that one or more might fail.

As the number of failures within an interval can only take whole number (integer) values, the Poisson distribution is a discrete distribution. Its probability density function for the number of failures x in an interval of length t is:

The Poisson distribution has the single parameter - it is called the average rate of the Poisson process. The mean of the distribution is t and the standard deviation is .

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