Calculating Poisson Probabilities (Edexcel International A Level (IAL) Maths: Statistics 2): Revision Note

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 ,...., n

• 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? 

• Most of the time the value of λ will be in the table for the 'Poisson Cumulative Distribution Function' in the formula booklet

• $\mathrm{P}(X\le x)$ is asking you to find the probabilities of all values up to and including x   

  • This will be the value in the row of x

• $P(X<x)$ 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

  • This will be the value in the row of x - 1

    • $\mathrm{P}(X\le x-1)$

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

    • Using above we get: $\mathrm{P}(X\ge x)=1-\mathrm{P}(X\le x-1)$

• $P(X>x)$ 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 $P(a\le X\le b)$ then calculate $\mathrm{P}(X\le b)-\mathrm{P}(X\le a-1)$

  • This is the same idea as for the binomial distribution

How do I calculate the cumulative probabilities for a Poisson distribution if λ is not in the table?

• In the unlikely case that λ is not in the table

  • 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

• 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