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

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Calculating Poisson Probabilities

Throughout this section we will use the random variable X tilde Po left parenthesis lambda right parenthesis  . 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
    • straight P left parenthesis X equals r right parenthesis equals p subscript r equals e to the power of negative lambda end exponent cross times fraction numerator lambda to the power of r over denominator r factorial end fraction 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
  • begin mathsize 16px style straight P left parenthesis X less or equal than x right parenthesis end style 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
  • begin mathsize 16px style P left parenthesis X less than x right parenthesis end style 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
      • begin mathsize 16px style straight P left parenthesis X less or equal than x minus 1 right parenthesis end style
  • begin mathsize 16px style straight P left parenthesis X greater or equal than x right parenthesis end style 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 begin mathsize 16px style P left parenthesis X greater or equal than x right parenthesis equals 1 minus P left parenthesis X less than x right parenthesis end style
      • Using above we get: begin mathsize 16px style straight P left parenthesis X greater or equal than x right parenthesis equals 1 minus straight P left parenthesis X less or equal than x minus 1 right parenthesis end style
  • begin mathsize 16px style P left parenthesis X greater than x right parenthesis end style 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 begin mathsize 16px style P left parenthesis X greater than x right parenthesis equals 1 minus P left parenthesis X less or equal than x right parenthesis end style as to calculate this
  • If calculating begin mathsize 16px style P left parenthesis a less or equal than X less or equal than b right parenthesis end style then calculate begin mathsize 16px style straight P left parenthesis X less or equal than b right parenthesis minus straight P left parenthesis X less or equal than a minus 1 right parenthesis end style
    • 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 begin mathsize 16px style P left parenthesis X less or equal than 3 right parenthesis equals e to the power of negative lambda end exponent cross times fraction numerator lambda to the power of 0 over denominator 0 factorial end fraction plus e to the power of negative lambda end exponent cross times fraction numerator lambda to the power of 1 over denominator 1 factorial end fraction plus e to the power of negative lambda end exponent cross times fraction numerator lambda squared over denominator 2 factorial end fraction plus e to the power of negative lambda end exponent cross times fraction numerator lambda cubed over denominator 3 factorial end fraction end style
    • Note that e to the power of negative lambda end exponent exists in every term and can be factorised out
    • Recall that begin mathsize 16px style lambda to the power of 0 equals 1 end style and 0! = 1
    • Recall also that begin mathsize 16px style lambda to the power of 1 equals lambda end style and 1! = 1
  • This calculation could be factorised and simplified to
    • begin mathsize 16px style P left parenthesis X less or equal than 3 right parenthesis equals e to the power of negative lambda end exponent open parentheses 1 plus lambda plus fraction numerator lambda squared over denominator 2 factorial end fraction plus fraction numerator lambda cubed over denominator 3 factorial end fraction close parentheses end style 
    • 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

Worked example

Xiao makes silly mistakes in his maths homework at a mean rate of 2 per page.

(a)
Define a suitable distribution to model the number of silly mistakes Xiao would make in a piece of homework that is five pages long.

(b)
Find the probability that in any random page of Xiao’s homework book there are
(i)
exactly three silly mistakes
(ii)
at most three silly mistakes
(iii)
more than three silly mistakes.
(a)
Define a suitable distribution to model the number of silly mistakes Xiao would make in a piece of homework that is five pages long.

1-2-2-calculating-poisson-probabilities-we-solution-part-1

(b)
Find the probability that in any random page of Xiao’s homework book there are 
(i)
exactly three silly mistakes 
(ii)
at most three silly mistakes 
(iii)
more than three silly mistakes.

1-2-2-calculating-poisson-probabilities-we-solution-part-2

Exam Tip

  • Look carefully at the given time or space interval to check if you need to change the mean before carrying out calculations.
    • Be prepared for this to change between question parts!

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