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Another important application of the theorem is that the binomial
and the Poisson distribution can be approximated, for ``large numbers'',
by a normal distribution. This is a general result, valid for
all distributions which have the *reproductive property
under the sum*. Distributions of this kind are the binomial,
the Poisson and the . Let us go into more detail:
**
**
- The reproductive property of the binomial states that if ,
, , are independent variables,
each following a binomial distribution of parameter and ,
then their sum
also follows a binomial distribution
with parameters
and . It is easy to be convinced
of this property without
any mathematics. Just think of what happens if one tosses bunches
of three, of five and of ten coins, and then one considers
the global result:
a binomial with a large can then always
be seen as a sum of many binomials with smaller . The
application of the central limit theorem is straightforward,
apart from deciding when the convergence is acceptable.
The parameters on which one has to base a judgment
are in this case and the
complementary quantity
. If they are
__both__
then the approximation starts to
be reasonable.
**
**
- The same argument holds for the Poisson distribution.
In this case the approximation starts to be reasonable
when
.

** Next:** Normal distribution of measurement
** Up:** Central limit theorem
** Previous:** Distribution of a sample
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Giulio D'Agostini
2003-05-15