2.1.1 The Poisson Distribution

What is a Poisson distribution?

• A Poisson distribution is a discrete probability distribution
• The discrete random variable  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  follows a Poisson distribution then it is denoted 
  •  is the mean number of trials
• The formula for the probability of r occurrences in a given interval is:
  •                 for r=0, 1, 2, .....,
  • 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

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

Throughout this section we will use the random variable   . For a Poisson distribution, the probability of a X taking a non-integer or negative value is always zero. Therefore any values mentioned in this section (other than λ) will be assumed to be non-negative integers.

Where does the formula for a Poisson distribution come from?

• The formula for calculating an individual Poisson probability is
  • for r = 0, 1, 2 ,....
• The derivation of the formula comes from the binomial distribution, however it is outside the scope of this syllabus and will not be proved here
  • Whilst the binomial distribution relies on knowing a fixed number of trials, the Poisson can allow for any number of trials within a time period
  • Only the mean number of occurrences of an event within the given period needs to be known

How do I calculate the cumulative probabilities for a Poisson distribution? 

• To find an individual probability use the formula 
• Most of the time you will be required to calculate cumulative Poisson probabilities rather than individual ones
  • Use the formula to find the individual probabilities and then add them up
  • Make sure you are confident working with inequalities for discrete values
  • Only integer values will be included so it is easiest to look at which integer values you should include within your calculation
  • Sometimes it is quicker to find the probabilities that are not being asked for and subtract from one
• is asking you to find the probabilities of all values up to and including x
  • This means all values that are at most x
  • Don’t forget to include P(X = 0)
• is asking you to find the probabilities of all values up to but not including x
  • This means all values that are less than x
  • Stop at x-1
• is asking you to find the probabilities of all values greater than and including x
  • This means all values that are at least x
  • As there is not a fixed number of trials this includes an infinite number of possibilities
  • To calculate this, use the identity 
• is asking you to find the probabilities of all values greater than but not including x
  • This means all values that are more than x
  • Rewrite  as to calculate this
• If calculating pay attention to whether the probability of a and b should be included in the calculation or not
  • For example, :
    • You want the integers 5 to 10

How can calculating probabilities for a Poisson distribution be made easier?

• Having to type a lot of calculations into your calculator can be time consuming and cause errors
• Consider the calculation 
  • Note that exists in every term and can be factorised out
  • Recall that  and 0! = 1
  • Recall also that  and 1! = 1
• This calculation could be factorised and simplified to
  •  
  • This is much simpler and easier to type into your calculator in exam conditions

How do I change the mean for a Poisson distribution?

• Sometimes the mean may be given for a different interval of time or space than that which you need to calculate the probability for
• A given value of λ can be adjusted to fit the necessary time period
  • For example if a football team score a mean of 2 goals an hour and we want to find the probability of them scoring a certain number of goals in a 90 minute game, then we would use the distribution X ~ Po(3)   
    • 90 = 1.5 (60) so use 1.5λ
• A very useful property of the Poisson distribution is that if X and Y are two independent Poisson distributions then their sums, X + Y is also a Poisson distribution
  • If X ~ Po(λ) and Y ~ Po(μ) then X + Y ~ Po(λ + μ)
  • Note that, for an integer value of a and b greater than 1, aX + bY no longer follows a Poisson distribution