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Enrique Treviño's 2021 Project

Exploring taking items out of an urn with different types of replacement. Suppose an urn has R red balls and W white balls for a total of N = R+W balls. Suppose that when you take out a white ball, you keep it with probability w (and return it otherwise), and if you take out a red ball, you keep it with probability r.  Fix a number n smaller than min(R,W). Take a sample of n balls from the urn one at a time with the replacement rules described above. Let X        be the number of white balls that are in the sample. When r=w=0, this is the classical "sampling with replacement" and X        corresponds to the binomial model, while r=w=1 is "sampling without replacement" and X       corresponds to the hypergeometric model. In their paper To replace or not to replace, Engbers and Hammett study X      . In this project we will work on some open questions regarding X       and will try to find nice expressions for the density, expectation and variance of X       .

(r,w)

(0,0)

(0,1)

(1,1)

(0,1)

(r,w)

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