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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)"

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

> For instance, n = 1000 and p = 0.001, then is 1000 is large?,
> is 0.001  is very small  or not too near 0 ?
> is np (1000 * 0.001 = 1) of moderate magnitude?
> Should I approximate this by possion dist or normal ?

> Would you please give me an idea when I should use possion/normal dist.
> to approximate binomial distribution.

There are some simple theorems about the accuracy, but these are conservative
for some purposes.  However, they are good guidelines.

The error in the Poisson approximation to the binomial is bounded by np^2,
where this is the sum of the absolute errors to the probabilities.  This can
be moderately improved for large variance.  In your example above, this would
make the error bound .001, and the actual error is about .000552.

The error in the central limit theorem has two main terms; a multiple of
the standardized third moment divided by the standard deviation, and a
bounded expression divided by the variance.  These expansions and error
bounds can be found in the standard works.  Since the variance is .999,
this approximation will be quite poor.

There are formulas to compute the binomial cdf by continued fractions 
from the probability of the endpoint.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
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