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

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
  • For a random variable that has the Poisson distribution you can write   
  • is the number of occurrences of the event in a particular interval
  • is the Poisson parameter
    • In fact,  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, , 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

What are the probabilities for the Poisson distribution?

• If , then  has the probability function:
  • 
• 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 also appear in the exam formula booklet
• It is possible for  to take any integer value greater than or equal to zero
  • I.e., there is no 'maximum possible value' for 
  • However each becomes closer and closer to zero as  becomes larger and larger

• Using the Maclaurin series of (along with ) gives

• • Then dividing both sides by  gives 

•  
  • 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  up or down by the same factor
    • For example, if the number of text messages received in an hour has the distribution
    • Then the number received in 3 hours has the  distribution
    • And the number received in 15 minutes has the  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 
  • Some calculators can only find 
    • The identities below will help in this case
  • Note that the values for  are also found in the tables in the exam formula booklet
    • But only for certain  values between 0.5 and 10

• As the Poisson distribution is for  you could rewrite all strict inequalities (< and >) as weak inequalities (≤ and ≥) using the following identities
  •  
    • For example,  means  so
  • 
    • For example,  means  so 
• You can reverse the sign of an inequality using
  • Be careful with which integer goes in the inequality
    • Listing the integers can help
  • For example,
    • Because  is everything except
• Note that
  • a Poisson random variable cannot take negative values
• Similarly,
  • is  take away 

What are the mean and variance of the Poisson distribution?

• If , then
  • The mean of is 

• • The variance of is  
• 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

What about the sum of two or more Poisson distributions?

• If  and are two independent Poisson variables,
  • then
    • 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
  • 
•  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