The Poisson Distribution (Edexcel A Level Further Maths: Further Statistics 1)

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Author

Roger

Last updated

12 September 2023

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 $Po (λ)$
- For a random variable that has the Poisson distribution you can write $X ~ Po (λ)$
- $X$ is the number of occurrences of the event in a particular interval
- $λ$ is the Poisson parameter
  - In fact, $λ$ 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, $X$ , 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

Examiner Tip

Replace the words "occurrences" or "events" with the context (e.g. "number of people arriving") when commenting on conditions and assumptions

Poisson Probabilities

What are the probabilities for the Poisson distribution?

If $X ~ Po (λ)$ , then $X$ has the probability function:
- $P (X = x) = e^{- λ} \frac{λ^{x}}{x!}, x = 0, 1, 2, 3, . . .$
It can be useful to know that formula
- But usually you will calculate Poisson probabilities using the stats functions on your calculator

- Cumulative Poisson probability tables for certain values of $λ$ also appear in the exam formula booklet
It is possible for $X$ to take any integer value greater than or equal to zero
- I.e., there is no 'maximum possible value' for $X$
- However each $P (X = x)$ becomes closer and closer to zero as $x$ becomes larger and larger

Using the Maclaurin series of $e^{λ}$ (along with $λ^{0} = 1$ ) gives

$e^{λ} = λ^{0} + \frac{λ^{1}}{1!} + \frac{λ^{2}}{2!} + \frac{λ^{3}}{3!} + . . . + \frac{λ^{r}}{r!} + . . .$

- Then dividing both sides by $e^{λ}$ gives

$1 = λ^{0} e^{- λ} + \frac{λ^{1} e^{- λ}}{1!} + \frac{λ^{2} e^{- λ}}{2!} + \frac{λ^{3} e^{- λ}}{3!} + . . . + \frac{λ^{r} e^{- λ}}{r!} + . . .$

- So the sum of all Poisson probabilities is equal to 1
  - This is a requirement of any probability distribution

What if I want to change the interval for a Poisson distribution?

It is possible to scale the interval of a Poisson distribution up or down
- You just need to scale the Poisson parameter $λ$ up or down by the same factor
  - For example, if the number of text messages received in an hour has the $Po (λ)$ distribution
  - Then the number received in 3 hours has the $Po (3 λ)$ distribution
  - And the number received in 15 minutes has the $Po (\frac{λ}{4})$ distribution
- This works the same for intervals in time and intervals in space
Remember the 'constant average rate' condition for using the Poisson distribution
- This is what assures that the scaling up and down here is valid

How do I calculate cumulative probabilities for a Poisson distribution?

You should have a calculator that can calculate cumulative Poisson probabilities
- Most calculators will find $P (a \leq X \leq b)$
- Some calculators can only find $P (X \leq b)$
  - The identities below will help in this case
- Note that the values for $P (X \leq b)$ are also found in the tables in the exam formula booklet
  - But only for certain $λ$ values between 0.5 and 10

As the Poisson distribution is for $X = 0, 1, 2, . . .$ you could rewrite all strict inequalities (< and >) as weak inequalities (≤ and ≥) using the following identities
- $P (X < x) = P (X \leq x - 1)$
  - For example, $X < 5$ means $0, 1, 2, 3, 4$ so $P (X < 5) = P (X \leq 4)$
- $P (X > x) = P (X \geq x + 1)$
  - For example, $X > 3$ means $4, 5, 6, 7 . . .$ so $P (X > 3) = P (X \geq 4)$
You can reverse the sign of an inequality using $1 - . . .$
- Be careful with which integer goes in the inequality
  - Listing the integers can help
- For example, $P (X \geq 4) = 1 - P (X \leq 3)$
  - Because $4, 5, 6, 7, . . .$ is everything except $0, 1, 2, 3$
Note that $P (X \leq b) = P (0 \leq X \leq b)$
- a Poisson random variable cannot take negative values
Similarly, $P (5 \leq X \leq 9) = P (X \leq 9) - P (X \leq 4)$
- $5, 6, 7, 8, 9$ is $0, 12, . . ., 9$ take away $0, 1, 2, 3, 4$

Examiner Tip

Be sure you know how to find individual and cumulative Poisson probabilities on your calculator

Worked example

Rachel has determined that pieces of rubbish along the route she walks to school occur at a rate of 2.5 per 100 metres. Given that a Poisson model is appropriate in this situation, find the probability that there will be:

exactly 3 pieces of rubbish in a space of 100 metres

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at least 1 piece of rubbish in a space of 50 metres

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no more than 10 pieces of rubbish in a space of 400 metres

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at least 15 but fewer than 30 pieces of rubbish in a space of 1 kilometre.

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Poisson Mean & Variance

What are the mean and variance of the Poisson distribution?

If $X ~ Po (λ)$ , then
- The mean of $X$ is $E (X) = λ$

- The variance of $X$ is $Var (X) = σ^{2} = λ$
Note that the mean and variance are the same for the Poisson distribution

How can the mean and variance of a sample indicate whether a Poisson model is appropriate?

The mean and variance being the same is a key property of the Poisson distribution
If you are given a sample of data and asked whether a Poisson model would be appropriate for modelling the data:
- Calculate the mean and variance for the sample
- If they are approximately equal, this suggests that a Poisson distribution may be a suitable model
  - Though always keep in mind any other Poisson conditions required (e.g. independence)
- If they are not approximately equal, then a Poisson distribution cannot be an appropriate model for the data

Examiner Tip

If given data from a sample, justifying why a Poisson model is (or isn't) appropriate almost always means showing that the sample's mean and variance are (or aren't) approximately equal

Worked example

A student counts the number of pieces of pineapple, $x$ , on each of 50 pineapple pizzas that she ordered for a school event. The results are summarised below.

$Σ x = 1259$ , $Σ x^{2} = 32964$

Calculate the mean and variance of the number of pieces of pineapple per pizza for the 50 pizzas.

poisson-mean-var-we-a

Explain how the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of pieces of pineapple on a pizza.

Sum of Poisson Distributions

What about the sum of two or more Poisson distributions?

If $X ~ Po (λ)$ and $Y ~ Po (μ)$ are two independent Poisson variables,
- then $X + Y ~ Po (λ + μ)$
  - So the sum of two Poisson variables is also a Poisson variable
  - And its mean is just the sum of the two means
This extends to n independent Poisson variables $X_{i} ~ Po (λ_{i})$
- $X_{1} + X_{2} + \dots + X_{n} ~ Po (λ_{1} + λ_{2} + \dots + λ_{n})$
But note that to add Poisson variables together
- they must all model events occurring over the same interval
  - e.g. over an interval of 5 minutes
  - or over an area of 5 square metres
- They do not have to model the same exact events
  - see the Worked Example

Examiner Tip

When asked to state an assumption you have made, It will usually be that the Poisson variables you are adding together are independent

Worked example

Pedestrians pass by Jovan's window at an average rate of 5.2 per hour. Cyclists pass by his window at an average rate of 3.8 every 30 minutes. Assuming that numbers of pedestrians and numbers of cyclists passing by Jovan's window may each be modelled by a Poisson distribution, find the probability that:

a total of exactly 15 pedestrians and cyclists will pass by Jovan's window in an hour

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