"Having introduced the idea of approximating the binomial distribution with two distributions, each of which is applicable in a different regime of the value of p, lets consider the case where p is small (p0.1). First, let us perform the substitution l =np. The binomial distribution then becomes,
Now, consider the case where n grows to infinity and p shrinks to zero. Hopefully you appreciate the utility of the substitution that we made above, since we can force n to grow and p to shrink such that l =np remains constant. This is nice since nothing in the above expression will "blow up" for large n and/or small p. In this limit we get,
(8) for k=0,1,2,3,...
This expression is the Poisson distribution, and is useful in the situations where the probability of an occurrence is small and the number of "trials" (n) is large. For example, we might consider the probability of k adverse reactions to a test drug in a given sample of the population or the probability of registering k complaints about a particular product in a 1-hour period or the probability of finding k point mutations in a given stretch of nucleotides. Though the Poisson distribution is essential to application and you will doubtless see it again, we will leave it now to discuss the other binomial-approximating continuous distribution. "