Poisson Approximations of Binomials (Edexcel A Level Further Maths): Revision Note

Poisson Approximations of Binomials

When can I use a Poisson distribution to approximate a binomial distribution?

• A binomial distribution can be approximated by a Poisson distribution  provided

  • n is large

  • p is small

  • There is no firm rule for what 'large' and 'small' mean here

    • is a good guide for 'large ' 

    • usually the value of should be

• The mean to use in the approximation can be calculated by:

  • 

  • This gives the Poisson the same mean as the binomial

  • Recall that for the binomial distribution

    • the mean is 

    • the variance is 

• If  is large but  is near to 1, consider modelling the number of failures, 

  • 

    • will be small

    • A Poisson approximation can then be used

• The Poisson distribution is derived from the binomial distribution by letting n become infinitely large and p become infinitely small