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A toy is randomly put in a given Cereal box as a promotional gift. There can be N different types of toys and each one can be of any type N (IID). (a) Find the expected number of cereal box one has to buy before she can have atleast one toy of each type, (b) If she has already collected m toys, what is the expected number of different toys she has collected.

Can someone explain how to model this problem using Indicator Random variables.

joriki
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Rahul
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  • (a) is the well-known Coupon collector's problem. You'll find solutions for it under that name across the Web, and in the duplicate thread. (b) is not a well-posed question. There's no obvious prior for the number of cereal boxes bought. You could turn it into a well-posed question by asking for the expected number of cereal boxes required to collect $m$ different toys. In that case, the solution is obtained immediately by applying the same approach as for the case $m=N$ addressed in (a). – joriki May 30 '16 at 16:55
  • Thanks joriki. The first part is clear. Though I am not sure how to approach part (b). – Rahul May 30 '16 at 16:58
  • You'd first need to clarify the question. Did you mean the number of cereal boxes required to collect $m$ different toys? If not, what did you mean? Unless I'm missing something, as currently phrased the question would require some prior over the number of boxes bought. Since there's no uniform prior over the natural numbers, it's not obvious which prior you may have intended to imply. – joriki May 30 '16 at 17:00
  • My bad. I have fixed it. – Rahul May 30 '16 at 17:06
  • That's a completely different question, which has been addressed in various forms in various threads on this site, e.g.: http://math.stackexchange.com/questions/28930, http://math.stackexchange.com/questions/879104, http://math.stackexchange.com/questions/545920. (The number of toys not collected is the number of empty bins.) – joriki May 30 '16 at 17:10

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