Coupon Collector Problem with Reset Button
We consider the following generalization of the classical coupon collector problem. We assume that, in addition to the initial collection of standard coupons, there is one more coupon that acts as a reset button, removing all coupons from the part of the collection that has already been drawn. For t...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/2/239 |
| _version_ | 1797343054943748096 |
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| author | Jelena Jocković Bojana Todić |
| author_facet | Jelena Jocković Bojana Todić |
| author_sort | Jelena Jocković |
| collection | DOAJ |
| description | We consider the following generalization of the classical coupon collector problem. We assume that, in addition to the initial collection of standard coupons, there is one more coupon that acts as a reset button, removing all coupons from the part of the collection that has already been drawn. For the case where standard coupons have unequal probabilities of being drawn, we obtain the distribution of the waiting time until the end of the collection process. For the case where standard coupons have equal probabilities, we derive a simple formula for the expected waiting time in terms of the beta function, and discuss the asymptotic properties of this expected waiting time, when the number of standard coupons tends toward infinity. |
| first_indexed | 2024-03-08T10:42:08Z |
| format | Article |
| id | doaj.art-429f56c4faa44838bb2c3d365f9703a5 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T10:42:08Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-429f56c4faa44838bb2c3d365f9703a52024-01-26T17:31:29ZengMDPI AGMathematics2227-73902024-01-0112223910.3390/math12020239Coupon Collector Problem with Reset ButtonJelena Jocković0Bojana Todić1Faculty of Mathematics, University of Belgrade, 11000 Belgrade, SerbiaFaculty of Mathematics, University of Belgrade, 11000 Belgrade, SerbiaWe consider the following generalization of the classical coupon collector problem. We assume that, in addition to the initial collection of standard coupons, there is one more coupon that acts as a reset button, removing all coupons from the part of the collection that has already been drawn. For the case where standard coupons have unequal probabilities of being drawn, we obtain the distribution of the waiting time until the end of the collection process. For the case where standard coupons have equal probabilities, we derive a simple formula for the expected waiting time in terms of the beta function, and discuss the asymptotic properties of this expected waiting time, when the number of standard coupons tends toward infinity.https://www.mdpi.com/2227-7390/12/2/239coupon collector problemreset couponexpected waiting timeMarkov chainbeta function |
| spellingShingle | Jelena Jocković Bojana Todić Coupon Collector Problem with Reset Button Mathematics coupon collector problem reset coupon expected waiting time Markov chain beta function |
| title | Coupon Collector Problem with Reset Button |
| title_full | Coupon Collector Problem with Reset Button |
| title_fullStr | Coupon Collector Problem with Reset Button |
| title_full_unstemmed | Coupon Collector Problem with Reset Button |
| title_short | Coupon Collector Problem with Reset Button |
| title_sort | coupon collector problem with reset button |
| topic | coupon collector problem reset coupon expected waiting time Markov chain beta function |
| url | https://www.mdpi.com/2227-7390/12/2/239 |
| work_keys_str_mv | AT jelenajockovic couponcollectorproblemwithresetbutton AT bojanatodic couponcollectorproblemwithresetbutton |