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

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 $\mathrm{Po}\left(\lambda \right)$

  • For a random variable that has the Poisson distribution you can write $X~\mathrm{Po}\left(\lambda \right)$  

  • $X$ is the number of occurrences of the event in a particular interval

  • $\lambda$ is the Poisson parameter

    • In fact, $\lambda$ 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

Poisson probabilities

What are the probabilities for the Poisson distribution?

• If $X~\mathrm{Po}\left(\lambda \right)$, then $X$ has the probability function:

  • $\mathrm{P}\left(X=x\right)={\mathrm{e}}^{-\lambda }\frac{{\lambda }^x}{x!},x=0,1,2,3,...$

• 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 $\lambda$ 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 $\mathrm{P}\left(X=x\right)$ becomes closer and closer to zero as $x$ becomes larger and larger

• Using the Maclaurin series of ${\mathrm{e}}^{\lambda }$ (along with ${\lambda }^0=1$) gives

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

• Then dividing both sides by ${\mathrm{e}}^{\lambda }$ gives 

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

•  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 $\lambda$ up or down by the same factor

    • For example, if the number of text messages received in an hour has the $\mathrm{Po}\left(\lambda \right)$ distribution

    • Then the number received in 3 hours has the $\mathrm{Po}\left(3\lambda \right)$ distribution

    • And the number received in 15 minutes has the $\mathrm{Po}\left(\frac{\lambda }{4}\right)$ 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 $\mathrm{P}(a\le X\le b)$

  • Some calculators can only find $\mathrm{P}(X\le b)$

    • The identities below will help in this case

  • Note that the values for $\mathrm{P}(X\le b)$ are also found in the tables in the exam formula booklet

    • But only for certain $\lambda$ 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

  • $\mathrm{P}(X<x)=\mathrm{P}(X\le x-1)$ 

    • For example, $X<5$ means $0,1,2,3,4$ so $\mathrm{P}\left(X<5\right)=\mathrm{P}\left(X\le 4\right)$

  • $\mathrm{P}(X>x)=\mathrm{P}(X\ge x+1)$

    • For example, $X>3$ means $4,5,6,7...$ so $\mathrm{P}\left(X>3\right)=\mathrm{P}\left(X\ge 4\right)$

• 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, $\mathrm{P}\left(X\ge 4\right)=1-\mathrm{P}\left(\mathrm{X}\le 3\right)$

    • Because $4,5,6,7,...$ is everything except $0,1,2,3$

• Note that $\mathrm{P}(X\le b)=\mathrm{P}(0\le X\le b)$

  • a Poisson random variable cannot take negative values

• Similarly, $\mathrm{P}\left(5\le X\le 9\right)=\mathrm{P}\left(X\le 9\right)-\mathrm{P}\left(X\le 4\right)$

  • $5,6,7,8,9$ is $0,12,...,9$ take away $0,1,2,3,4$

Poisson mean & variance

What are the mean and variance of the Poisson distribution?

• If $X~\mathrm{Po}\left(\lambda \right)$, then

  • The mean of $X$ is  $\mathrm{E}\left(X\right)=\lambda$

  • The variance of $X$ is  $\mathrm{Var}\left(X\right)={\sigma }^{\mathit{2}}\mathit{=}\lambda$

• 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

Sum of Poisson distributions

What about the sum of two or more Poisson distributions?

• If $X~\mathrm{Po}\left(\lambda \right)$ and $Y~\mathrm{Po}\left(\mu \right)$ are two independent Poisson variables,

  • then $\mathit{X}\mathbf{+}\mathit{Y}\mathbf{~}\mathbf{Po}\mathbf{(}\mathit{\lambda }\mathbf{+}\mathit{\mu }\mathbf{)}$

    • 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 $X_i~\mathrm{Po}\left({\lambda }_i\right)$

  • $X_1+X_2+ \dots +X_n~\mathrm{Po}\left({\lambda }_1+{\lambda }_2+ \dots +{\lambda }_n\right)$

•  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