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

Calculating Poisson 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 ,...., 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

•  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

• 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

    • 

• 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 

    • Using above we get: 

•  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 then calculate 

  • 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 

  • 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