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

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Amber

Last updated

19 December 2023

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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 $X ~ Po (λ)$
- λ is the mean number of occurrences of the event
The formula for the probability of r occurrences in a given interval is:
- $P (X = r) = e^{- λ} \times \frac{λ^{r}}{r!}$ for r=0, 1, 2, ...,n
- e is the constant 2.718…
- $r! = r (r - 1) (r - 2) . . . . 2 \times 1$

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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