Poisson approximation to the binomial distribution

Introduction to Poisson approximation to the Binomial Distribution

  • The Poisson Distribution is a discrete probability distribution expressing the probability of a given number of events occurring in a fixed interval of time or space.
  • The Binomial Distribution is also a discrete distribution that describes the outcome of ‘n’ independent trials in an experiment. Each trial is assumed to result in a success with probability ‘p’ or a failure with probability ‘1-p’.
  • The Poisson approximation to the Binomial distribution is used when we have a binomial distribution where ‘n’ is large and ‘p’ is small, such that the product ‘np’ is moderate. The random variable ‘X’ is then approximated to a Poisson distribution where the mean ‘λ’ is equal to ‘np’.

Principles of the Approximation

  • Under suitable conditions, the Poisson distribution can provide an excellent approximation to the binomial distribution.
  • The approximation works best when ‘n’ is large, and ‘p’ is small such that ‘np’ is between 5 to 10.
  • If ‘np’ is greater than 10, it is generally better to use a normal approximation to the binomial.

Applying the Poisson Approximation

  • If the conditions are met (‘n’ large, ‘p’ small, ‘np’ moderate), the binomial distribution can be approximated by the Poisson with parameter λ=np.
  • For example, if you have a binomial distribution with ‘n’ = 100 and ‘p’ = 0.02, you can use a Poisson approximation with ‘λ’ = 100 x 0.02 = 2.
  • Use the formula for the Poisson distribution: P(X=k) = λ^k * e^(-λ) / k!, where e is the base of natural logarithms (approx. 2.72) and ! denotes a factorial.

Impact and Limitations of the Approximation

  • The Poisson approximation is extremely useful in real-world applications where events are rare, such as the occurrence of rare diseases or phone calls received at a call center.
  • However, it’s important to remember that the approximation is just that - an approximation. There may still be minor discrepancies between the actual binomial distribution and the approximated Poisson distribution.
  • Keep in mind that, like all statistical models, the Poisson approximation is only as good as the assumptions on which it’s based and does not replace the more accurate binomial distribution when ‘n’ is small or ‘p’ is not sufficiently small.