Exam Questions - Poisson approximation to the binomial distribution

Exam Questions - Poisson approximation to the binomial distribution

Understanding Poisson Approximation to the Binomial Distribution

  • A Poisson distribution is a discrete probability distribution often used for modelling unpredictable events that occur over a fixed interval of time or space.
  • Poisson approximation is applied to the binomial distribution when the number of trials (n) is large, while the probability of success (p) is small, resulting in np being approximately equal to the mean (λ).
  • For binomial distribution, the number of trials (n) and the probability of success (p) on each trial are known. But realistically, when n increases and p decreases such that np remains constant, it becomes challenging to compute binomial probabilities. Hence, we use the Poisson approximation.

Using Poisson Approximation

  • The approximation is based on the formula P(X = r) = (λ^r * e^-λ) / r!, where r is the number of successes, λ = np and e^-λ is a constant.
  • This formula is used to calculate Poisson probabilities when the binomial conditions are unsuitable.
  • Poisson approximation simplifies the calculation of binomial probabilities.

Applications of Poisson Approximation

  • Poisson approximation is used for estimating the number of events over a specific interval of time or space.
  • It is extensively applied in various fields, including telecommunication, biology, astronomy, and queuing systems to model random point processes.
  • This approximation helps in modelling scenarios where an event can occur a large number of times but each occurrence is unlikely.

Interpreting Poisson Approximation Results

  • Just like any other statistical model, the interpretation of the results obtained via Poisson approximation depends on the context.
  • The probabilities obtained can help us predict the likelihood of a number of events happening over a time or space interval.
  • A higher Poisson probability implies the event is likely to happen according to the defined conditions.

Poisson Approximation in Practice

  • Remember to check whether conditions are suitable for Poisson approximation before carrying out calculations.
  • The Poisson distribution has only one parameter, i.e., its mean, simplifying the calculation as compared to binomial distributions.
  • Make sure to understand the context of the problem. Real-life scenarios play a crucial role in determining the use of approximations in statistics.