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

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Amber

Expertise

Maths

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
• 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…

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

Worked example

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  to follow a Poisson distribution.

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.

Exam 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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