Prompt Title: Poisson Approximation to Binomial

The Poisson distribution is often used as an approximation to the binomial distribution in situations where the number of trials is very large (n → ∞) and the probability of success is very small (p → 0), but the product np remains constant.

In such situations, the binomial distribution can become computationally cumbersome to work with because there are a large number of possible outcomes to consider, each with a very small probability. The Poisson distribution, on the other hand, is simpler to work with because it only has one parameter, λ, which represents the average number of occurrences per unit of time or space.

When np is constant and p is small, the Poisson distribution with parameter λ = np can be used as an approximation to the binomial distribution. This is because the Poisson distribution has a similar shape to the binomial distribution, but is easier to calculate and work with.

The approximation works best when np is less than or equal to 10 and p is less than or equal to 0.1. When np is greater than 10 or p is greater than 0.1, the normal distribution can be used as an even better approximation to the binomial distribution.