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December 14th, 2010, 03:04 AM   #1
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Number of unique after n samples

Consider an urn containing B balls.

What is the expression for the expected number of unique balls drawn after n samples, if sampling is independent and with replacement?

This one has me stumped, even though it appears simple.

Can anyone on here crack it?

Regards,

BigBugBuzz
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December 14th, 2010, 02:09 PM   #2
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Re: Number of unique after n samples

The question is a little unclear. Are the balls all the same or are they marked different? Does unique mean not picking a ball after it has already been picked and put back?
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December 15th, 2010, 03:56 AM   #3
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Re: Number of unique after n samples

Quote:
Originally Posted by mathman
The question is a little unclear. Are the balls all the same or are they marked different? Does unique mean not picking a ball after it has already been picked and put back?
Hi Mathman,

Let's say the balls have an ID form 1 to B.

As an example of number of unique, consider that n = 3 and that the following ID's were drawn 3, 5, 5.
Here the number of unique is 2, because the ID of 5 was drawn twice, but should only be counted once.

There is noting that prevents an ID being part of a given sample many times (because sampling is done with replacement), but it is of course unlikely for large values of B.

I hope this helps.
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December 15th, 2010, 02:25 PM   #4
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Re: Number of unique after n samples

You are dealing with a multinomial distribution. I suggest you look it up (google - Wikipedia, etc.).
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