the-poisson-distribution

Conditions for Poisson models

What is the Poisson distribution?

• The Poisson distribution is used to model events that occur randomly within an interval

  • This could be an interval in time

    • For example the number of calls received by a call centre per hour

  • Or an interval in space

    • For example, how many flowers of a particular kind are found per square metre of land

• The notation for the Poisson distribution is

  • For a random variable that has the Poisson distribution you can write

  • is the number of occurrences of the event in a particular interval

  • is the Poisson parameter

    • In fact,  is both the mean and the variance of the distribution 

What are the conditions for using a Poisson model?

• A Poisson distribution can be used to model the number of times, , that a specified event occurs within a particular interval of time or space

• In order for a Poisson distribution to be an appropriate model, the following conditions must all be satisfied:

  • The events must occur independently

  • The events must occur singly (in space or time)

    • Two (or more) events cannot happen at exactly the same time

  • The events must occur at a constant average rate

When might the conditions not be satisfied?

• If asked to criticise a Poisson model, you may be able to question whether occurrences of the event are really independent, happening singly or at a constant average rate

  • For example, when recording the number of people entering a restaurant in a given time interval

    • People entering may not be independent (they could be invited in by others they know)

    • People may not be entering singly (they could be entering at the same time in a group)

    • People entering may not be at a constant rate (there may be more at dinner time but fewer in the afternoon)

  • In order to proceed using the model, you would have to assume that the occurrences are independent, happen singly and at a constant average rate

Poisson probabilities

What are the probabilities for the Poisson distribution?

• If , then  has the probability function:

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