🚀 go-pugleaf

In article <1992Oct7.025857.28031@news2.cis.umn.edu> lee@hecto.cs.umn.edu (YoungJun Lee) writes:
>
> The stat. book I have just says that
>  "Binomial distribution with n trials and success probability p can be
>   approximated by possion distribution(when n is large, p is very small, and
>   np is of moderate magnitude" and normal distribution(when n is large and
>   p is not too near 0 or 1)"

For the Poisson approximation, it is much more important for p to be small
than it is for n to be large. The approximation is quite acceptable for n = 1
with p <= 0.01. This can be seen by looking at the relative error in the
variance; it involves only p.

With n = 30, a p of 0.05 or less is usually considered satisfactory.

> What's large n ? 
> What's "the very small p" or "not too near 0 or 1"?
> What's np of moderate magnitude ?

For the gaussian approximation, we were taught that the tails of the
approximating distribution should be well within the limits of 0 and n. Thus,
np - 5 (npq)**0.5 should be greater than zero, and np + 5(npq)**0.5 should
be less than n, where q = 1 - p. In other words, the mean should be at least
five standard deviations from both zero and n. Again, this is a rule of
thumb. For p = 0.5 (the best case), n should be at least 25. Smaller or
larger values of p will require a larger n to reduce the coefficient of
variation.