scholarly journals A coupon collector's problem with bonuses

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Toshio Nakata ◽  
Izumi Kubo
Keyword(s):  
Normal Distribution ◽  
Exponential Distribution ◽  
Gamma Distribution ◽  
Exact Distribution ◽  
Double Exponential Distribution ◽  
Coupon Collector’S Problem ◽  
Double Exponential ◽  
Different Types ◽  
International Audience ◽  
Coupon Collector's Problem

International audience 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.

Extremes for the minimal spanning tree on normally distributed points

1998 ◽  
Vol 30 (03) ◽  
pp. 628-639 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Normal Distribution ◽  
Exponential Distribution ◽  
Spanning Tree ◽  
Dimensional Space ◽  
Edge Length ◽  
Nearest Neighbour ◽  
Minimal Spanning Tree ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
Union Of Balls

Let n points be placed independently in ν-dimensional space according to the standard ν-dimensional normal distribution. Let M n be the longest edge-length of the minimal spanning tree on these points; equivalently let M n be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)1/2 M n - b n converges weakly to the Gumbel (double exponential) distribution, where b n are explicit constants with b n ~ (ν - 1)log log n. We also show the same result holds if M n is the longest edge-length for the nearest neighbour graph on the points.

Extremes for the minimal spanning tree on normally distributed points

1998 ◽  
Vol 30 (3) ◽  
pp. 628-639 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Normal Distribution ◽  
Exponential Distribution ◽  
Spanning Tree ◽  
Dimensional Space ◽  
Edge Length ◽  
Nearest Neighbour ◽  
Minimal Spanning Tree ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
Union Of Balls

Let n points be placed independently in ν-dimensional space according to the standard ν-dimensional normal distribution. Let Mn be the longest edge-length of the minimal spanning tree on these points; equivalently let Mn be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)1/2Mn - bn converges weakly to the Gumbel (double exponential) distribution, where bn are explicit constants with bn ~ (ν - 1)log log n. We also show the same result holds if Mn is the longest edge-length for the nearest neighbour graph on the points.

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