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A coupon collector's problem with bonuses
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities (2006).

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In this article, we study a variant of the coupon collector's problem introducing a notion of a \emphbonus. Suppose that there are c different types of coupons made up of bonus coupons and ordinary coupons, and that a collector gets every coupon with probability 1/c each day. Moreover suppose that every time he gets a bonus coupon he immediately obtains one more coupon. Under this setting, we consider the number of days he needs to collect in order to have at least one of each type. We then give not only the expectation but also the exact distribution represented by a gamma distribution. Moreover we investigate their limits as the Gumbel (double exponential) distribution and the Gauss (normal) distribution. 
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     author = {Nakata, Toshio and Kubo, Izumi},
     title = {A coupon collector's problem with bonuses},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities},
     year = {2006},
     doi = {10.46298/dmtcs.3515},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3515/}
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Nakata, Toshio; Kubo, Izumi. A coupon collector's problem with bonuses. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities (2006). doi : 10.46298/dmtcs.3515. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3515/

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