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Q: Continuous hypergeometric distribution ( No Answer,   0 Comments )
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Subject: Continuous hypergeometric distribution
Category: Science > Math
Asked by: marsal-ga
List Price: $30.00
Posted: 28 Aug 2003 04:38 PDT
Expires: 03 Sep 2003 08:33 PDT
Question ID: 249586
I would like to know the probability distribution of the following
problem:

Suppose I have an urn with N1 ones and N2 zeros. I then pick very
small fractions from this urn so that I eventually have accumulated n
units. (The picks are made without replacement.)
If I would call the sum that I have accumulated x, then I would like
to know the probability distribution of x/n.

I have already figured out the following:
If I would simply pick whole numbers, then the solution would be the
hypergeometric distribution.
Furthermore, if I would let the urn contain an infinite amount of ones
and zeros, but with a proportion p being ones, then I could use the
binomial distribution.
If I picked fractions from this infinite amount of ones and zeros,
instead of whole numbers, I would apparently get the beta
distribution.

Request for Question Clarification by mathtalk-ga on 02 Sep 2003 19:03 PDT
Hi, marsal-ga:

The Subject title of your question suggests that you are looking for
an answer that involves a continuous distribution, although the actual
distribution of the "statistic" x/n (a sample mean) is obviously a
discrete distribution for any fixed value of n.

In practice the discrete distribution is often modelled for this sort
of experiment (a Bernoulli trial) as if it were the continuous normal
distribution.  One rule of thumb often cited is for n > 25, but in
reality the quality of this approximation depends on the balance
between N1 and N2.  The greater the "imbalance" between the two
varieties, the greater the sample size n that is required for the
normal distribution to be an adequate approximation.

I can provide a more detailed analysis along these lines, but it seems
possible to me that a clarification of the question on your part might
lead to a deep understanding of what you'd like to know.

regards, mathtalk-ga

Clarification of Question by marsal-ga on 03 Sep 2003 01:17 PDT
Hi mathtalk-ga,

Thanks for your message.

Actually, since I pick fractions (and let them go to zero, which I
didn't write previously) from the urn, I think the distribution will
be continuous.

For example, if I would pick fractions of tenths, then two picks (i.e
n=0.2) could lead to x/n being 0, 0.5, or 1.

If I let n be constant, but decrease the size of the fractions to
twentieths, then I must make four picks to fill the "group". This
could lead to x/n being 0, 0.25, 0.5, 0.75, or 1.

The idea is then that the fraction goes to zero, which I think would
imply that the distribution becomes continuous.

I hope this is correct. I'm sorry I was a bit unclear in the original
question.

When it comes to approximating the distribution with the normal
distribution, it would unfortunately not help me. What really
interests me with this distribution are in fact the cases when it is
not normally distributed. (i.e. n<25 and/or large imbalance between N1
and N2)

Thanks again for your help.
marsal-ga
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