The Poisson Distribution (Edexcel International A Level Maths): Revision Note

A Poisson distribution is a discrete probability distribution

The discrete random variable X follows a Poisson distribution if it counts the number of events that occur at random in a given time or space

For a Poisson distribution to be valid it must satisfy the following properties:

• Events occur singly and at random in a given interval of time or space

• The mean number of occurrences in the given interval(λ)  is known and finite

  • λ has to be positive but does not have to be an integer

• Each occurrence is independent of the other occurrences

If X follows a Poisson distribution then it is denoted 

• λ is the mean number of occurrences of the event

The formula for the probability of r occurrences in a given interval is:

• for r=0, 1, 2, ...,n

• e is the constant 2.718…

• 

The mean and variance of a Poisson distribution are roughly equal

The distribution can be represented visually using a vertical line graph

• If λ is close to 0 then the graph has a tail to the right (positive skew)

• If λ is at least 5 then the graph is roughly symmetrical

The Poisson distribution becomes more symmetrical as the value of the mean (λ) increases

Find the mean and variance and check that they are roughly equal

• You may have to change the mean depending on the given time/space interval

Make sure you clearly state what your random variable is

• For example, let X be the number of typing errors per page in an academic article

Identify what probability you are looking for

Anything that occurs singly and randomly in a given interval of time or space and satisfies the conditions

For example, let X  be the random variable 'the number of emails that arrive into your inbox per day'

• There is a given interval of a day, this is an example of an interval of time

• We can assume the emails arrive into your inbox at random

• We can assume each email is independent of the other emails

  • This is something that you would have to consider before using the Poisson distribution as a model

• If you know the mean number of emails per day a Poisson distribution can be used

Sometimes the given interval will be for space

• For example, the number of daisies that exist on a square metre of grass

• look carefully at the units given as you may have to change them when calculating probabilities