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The first part of the question (Finding expected number of distinct values selected from a set of integers) has been answered a few times.

Can someone provide a solution if we're drawing from a binomial distribution instead of a uniform distribution?

In other words, let X1,X2,X3,…,Xk be independent Binomial (n,p) random variables. What is the expected number of distinct values in the set {X1,…,Xk}?

isthisthat
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  • What exactly do you mean by "drawing from a binomial distribution"? Drawing is something that takes place on base of a distribution. If it happens on base of uniform distribution then all objects have equal chances to be drawn. What are the objects here? – drhab May 05 '17 at 13:10
  • Let $X_1,X_2,X_3,\ldots, X_k$ are independent Binomial $(n,p)$ random variables. Do you ask about the number of distinct values in the set ${X_1,\ldots,X_k}$? – NCh May 05 '17 at 14:12
  • @NCh yes that's exactly what I mean! Thanks for the elegant formulation, I've updated the definition. – isthisthat May 08 '17 at 11:01
  • This is the "coupon collector's problem" with unequal probabilities. See the remarks on the general case at the bottom of https://en.wikipedia.org/wiki/Coupon_collector%27s_problem and the citation to a paper of Flajolet, although it may not be that easy to extract an answer from the formula there. – Michael Lugo May 08 '17 at 13:39
  • I have a feeling (from empirical evidence) that the number of distinct values in the set {X1,…,Xk} follows a geometric distribution but cannot demonstrate it mathematically. Anyone able to help? – isthisthat May 09 '17 at 11:43

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