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International A LevelMaths: Statistics 2EdexcelRevision Notes1. Statistical Distributions1.2 Poisson Distribution1.2.1 The Poisson Distribution

The Poisson Distribution (Edexcel International A Level Maths: Statistics 2)

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Properties of Poisson Distribution

What is a Poisson distribution?

  • 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 X tilde Po left parenthesis lambda right parenthesis
    • λ is the mean number of occurrences of the event
  • The formula for the probability of r occurrences in a given interval is:
    • P left parenthesis X equals r right parenthesis equals e to the power of negative lambda end exponent cross times fraction numerator lambda to the power of r over denominator r factorial end fraction for r=0, 1, 2, ...,n
    • e is the constant 2.718…
    • r factorial equals r open parentheses r minus 1 close parentheses open parentheses r minus 2 close parentheses....2 cross times 1

What are the important properties of a Poisson distribution?

  • 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

2-1-1-poisson-distribution-diagram-1

Worked example

 X is the random variable ‘The number of cars that pass a traffic camera per day’. State the conditions that would need to be met for X to follow a Poisson distribution.

2-1-1-the-poisson-distribution-we-solution-1

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Modelling with Poisson Distribution

How do I set up a Poisson model?

  • 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

What can be modelled using a Poisson distribution?

  • 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

Worked example

State, with reasons, whether the following can be modelled using a Poisson distribution and if so write the distribution.

(i)
Faults occur in a length of cloth at a mean rate of 2 per metre.

 

(ii)
On average 4% of a certain population has green eyes.

 

(iii)
An emergency service company receives, on average, 15 calls per hour.

2-1-1-modelling-with-a-poisson-we-solution-2

Examiner Tip

  • If you are asked to criticise a Poisson model always consider whether the trials are independent, this is usually the one that stops a variable from following a Poisson distribution!

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    Amber

    Author: Amber

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    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.