Continuing the discussion from Constraining Output of Multinomial Distribution:

Over time, I have realized that the Wallenius non-central hyoergeometric distribution is not the correct distribution for my needs. The reason is: I do not believe the probability of selection depends upon the number of items available.

An example that I constructed that enumerates the type of problem I am trying to solve:

Assume we have `c`

different t-shirts, each with a different probability of being purchased, `p_i, i=1...c.`

For each t-shirt, we have `m_i, i=1...c`

copies of it.

Now, a customer is shown all of the t-shirts we have in stock (that is, each t-shirt where `m_i > 0`

). They purchase one according to the probabilities of the shirts that are available. Then this t-shirt is selected, `m_i`

is decremented by one and the next customer is shown the t-shirts.

This example that I listed seems to be a more â€ścategorical sampling at each realizationâ€ť where the probabilities only change if we â€śstick outâ€ť of an item.

Does this correspond to a Fisher non-central hypergeometric distribution? If not, does anyone know of a non-central hypergeometric distribution that satisfies this type of sampling?

If so, and `c`

is large, does anyone have insight as to how to compute the denominator (the `P_0`

) in the Fisher distribution?