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I would like to know if there's a known distribution for a multinomial sampling with limited bin size, or equivalently without replacement. The situation would be that I have $N$ bags of candy holding $k_i, i\in 1..N$ candies within them. Assuming I eat candies randomly from these bags with fixed equal probability $p$, discarding empty bags once I've consumed the last candy, is there a well behaved distribution for the number of candies I've eaten from each bag? Failing that, is there a probability for bag $n$ being empty after $x$ trials?

shians
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1 Answers1

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Maybe the multivariate hypergeometric distribution fits your example. See Wikipedia's article on this distribution.

cquina
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    I failed to say so in the original question but it's different in the way that I believe the multivariate hypergeometric weights the probabilities of each bin based on the objects remaining inside. I'm looking for something where the bin probabilities are fixed like the multinomial but can run out. – shians Jan 02 '18 at 03:51
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    @shians: There is no such thing; the very nature of sampling without replacement is that it should reduce the subsequent probability of drawing the same outcome. That necessitates that the probabilities are not fixed. – Ben Aug 07 '19 at 22:24
  • @shians One relevant keyword is weighted reservoir sampling (see Eq. (1)). The setting is slightly different, where a subset of unique items are sampled from those categories given the probability of each category. – Cuong Mar 21 '24 at 17:04