The Poisson Distribution (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Revision Note

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 $X~\mathrm{Po}\left(\lambda \right)$

  • $\lambda$ is the mean number of trials

• The formula for the probability of r occurrences in a given interval is:

  • $P(X=r)=e^{-\lambda }\times \frac{{\lambda }^r}{r!}$                for r=0, 1, 2, .....,$n$

  • e is the constant 2.718…

  • $r!=r\left(r-1\right)\left(r-2\right)....2\times 1$

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

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

Calculating Poisson Probabilities

Throughout this section we will use the random variable $X~\mathrm{Po}\left(\lambda \right)$  . 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

  • $\mathrm{P}(X=r)=p_r=e^{-\lambda }\times \frac{{\lambda }^r}{r!}$ 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 $p_r=e^{-\lambda }\times \frac{{\lambda }^r}{r!}$

• 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

• $\mathrm{P}(X\le x)$ 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)

• $P(X<x)$ 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

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

• $P(X>x)$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 $P(X>x)=1-P(X\le x)$ as to calculate this

• If calculating $P(a\le X\le b)$pay attention to whether the probability of a and b should be included in the calculation or not

  • For example, $P(4<X\le 10)$ :

    • 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 $P(X\le 3)=e^{-\lambda }\times \frac{{\lambda }^0}{0!}+e^{-\lambda }\times \frac{{\lambda }^1}{1!}+e^{-\lambda }\times \frac{{\lambda }^2}{2!}+e^{-\lambda }\times \frac{{\lambda }^3}{3!}$

  • Note that $e^{-\lambda }$exists in every term and can be factorised out

  • Recall that ${\lambda }^0=1$ and 0! = 1

  • Recall also that ${\lambda }^1=\lambda$ and 1! = 1

• This calculation could be factorised and simplified to

  • $P(X\le 3)=e^{-\lambda }\left(1+\lambda +\frac{{\lambda }^2}{2!}+\frac{{\lambda }^3}{3!}\right)$ 

  • 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