On using one distribution to approximate another Every basic text about different statistical distributions make notions about when one distribution can be approximated using another one. For example, we are told that the binomial distribution with sufficiently large n and sufficiently small p can be approximated using Poisson distribution. My question is why? Why is this interesting? What do we gain by knowing that a certain process that is binomial in theory can be approximated by Poisson?
 A: Here's a couple of situations:
1) A problem I have had recently: I have a sum of many (several dozen) binomial$(n_i,p_i)$, with large $n_i$ (hundreds to thousands) and very small $p_i$ (in the general ballpark of $1/n_i$, perhaps typically between $0.1n_i$ and $10n_i$). The answers you get using the Poisson approximation to the binomial typically have a good deal more accuracy than required, so computing the convolution numerically is fortunately unnecessary (it's not particularly onerous, but less convenient, and not as easy to explain to people already aware of the existence of the Poisson approximation, but who have no experience of numerical convolution). 
2) I saw this one come up as a real problem quite recently: I have an average of $k$ independent variates, each hypergeometric. What's the probability that the average is less than $c$? If I already know that in some situations the normal is a reasonable approximation to the hypergeometric, I can exploit that immediately to answer the question (and for the values involved, the approximation is excellent).
