The Poisson Distribution - Research Paper Example

Poisson distribution has a number of important uses as will be shown later. When the binomial distribution for probabilities of success (p) is close to zero (0), say 0.01, it would be too time consuming to calculate, especially when N is large them the Poisson distribution becomes useful. According to Mason and Lind, (1999 p242): “The distribution of probabilities would become more and more skewed as the probability of success became smaller. The limiting form of the binomial distribution where the probability of success is very small and N is large is called the Poisson probability distribution.

” In general, the Poisson distribution will provide a good approximation to the binomial probabilities when N ≥ 20 and p ≤ 0.05. When N ≥ 100 and Np ≤ 10, the approximation will generally be excellent” (Miller and Miller, 1999 p187). Spiegel and Stephens, (1999 p158) also point to the relationship between the Binomial and Poisson distributions. “In the Binomial distribution, if N is large, while the probability p of the occurrence of an event is close to zero (0), so that q = 1 – p is close to 1, the event is called a rare event.

An event is rare if the number of trials is at least 50 (N ≥ 50) while Np is less than 5. In such case the binomial distribution is very closely approximated by the Poisson with λ = Np.” This is indicated by comparing the Poisson distribution table above with the Binomial distribution table shown immediately below. Spiegel and Stephens, (1999 p158) also indicated that: “Since there is a relationship between the binomial and normal distribution, it follows that there is also a relation between the Poisson and normal distributions.

It can in fact be shown that the Poisson distribution approaches a normal distribution with standardized variable (X – λ)/√λ as λ increases