Poisson Approximation to B(n, p) (AS)

The Poisson distribution is often used to approximate the binomial distribution, B(n, p), in specific situations. This technique is known as the Poisson approximation to the Binomial distribution.
The conditions under which the Poisson distribution can be used for approximation are usually when the number of trials (n) is large, and the success probability (p) is small.
The Poisson approximation is particularly useful when dealing with large values of n, where calculating binomial probabilities can be computationally demanding or practically infeasible.
The mean of the Poisson distribution used to approximate a binomial distribution B(n, p) is np or λ = np (λ is the symbol commonly used for the mean of a Poisson distribution). Therefore, when using the Poisson distribution to approximate the binomial distribution, set λ = np.
Remember that a key feature of the Poisson distribution is that the mean is equal to the variance. This can be a useful check to help confirm if the approximation is reasonable in a particular setting.
The Poisson approximation becomes more accurate as n increases and p decreases, ideally, np should be less than or equal to 5 for the approximation to provide a good estimate.
After applying the approximation, normal calculations for a Poisson distribution can be used to find probabilities. The formula for the Poisson distribution is P(X = x) = (λ^x e^-λ )/x!, where λ is the mean, e is a constant approximately equal to 2.71828 (Euler’s number), and x is the specific value for which you are calculating probability.
The Poisson approximation is utilised in a variety of fields, such as quality control, insurance, traffic flow analysis, and many others involving rare events over a large number of trials.
Bear in mind that the Poisson distribution only approximates the binomial distribution; they are not the same. Therefore, differences between results from the actual binomial distribution and the Poisson approximation can occur.
It’s beneficial to practise different problems with the Poisson approximation to the binomial distribution to get acquainted with its uses and quirks. Use past paper questions or problems from revision guides to ensure you are confident in applying this technique.
The concept of the Poisson approximation to B(n, p) might appear in questions involving hypothesis testing, expectation and variance, and probability distributions, among other topics, so it’s crucial to understand it thoroughly.