Answer to Question #198944 in Statistics and Probability for Lerato Mosupa

Question

Which one of the following probability distributions is used for determining whether or not a population has a Poisson probability distribution?

Answer 1

The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed.

Therefore, it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if $n ≥ 100\ and \ np ≤ 10.$

For example:

The number of mutations in a given sequence of DNA —the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution, that is

$X$ ~ $B(n,p)$

In such cases n is very large and p is very small (and so the expectation np is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution.

$X$ ~$Pois(np)$