Coupon Collector Research Articles

93.16 The coupon collector’s problem with unequal probabilities

93.16 The coupon collector’s problem with unequal probabilities

An Architectural Framework for Accurate Characterization of Network Traffic

In networks carrying large volume of traffic, accurate traffic characterization is necessary for understanding the dynamics and patterns of network resource usage. Previous approaches to flow characterization are based on random sampling of the packets (e.g., Cisco's NetFlow) or inferring characteristics solely based on long lived flows (LLFs) or on lossy data structures (e.g., bloom filters, hash tables). However, none of these approaches takes into account the heavy-tailed nature of the Internet traffic and separates the estimation algorithm from the flow measurement architecture.In this paper, we propose an alternate approach to traffic characterization by closely linking the flow measurement architecture with the estimation algorithm. Our measurement framework stores complete information related to short lived flows (SLFs) while collecting partial information related to LLFs. For real-time separation of LLFs and SLFs, we propose a novel algorithm based on typical sequences from information theory. The distribution (pdf) and sample space of the underlying traffic is estimated using the non-parametric Parzen window technique and likelihood function defined over the Coupon collector problem. We validate the accuracy and performance of our estimation technique using traffic traces from the internal LAN in our laboratory and from National Library for Applied Network Research (NLANR).

Using Theorem Proving to Verify Expectation and Variance for Discrete Random Variables

Statistical quantities, such as expectation (mean) and variance, play a vital role in the present age probabilistic analysis. In this paper, we present some formalization of expectation theory that can be used to verify the expectation and variance characteristics of discrete random variables within the HOL theorem prover. The motivation behind this is the ability to perform error free probabilistic analysis, which in turn can be very useful for the performance and reliability analysis of systems used in safety-critical domains, such as space travel, medicine and military. We first present a formal definition of expectation of a function of a discrete random variable. Building upon this definition, we formalize the mathematical concept of variance and verify some classical properties of expectation and variance in HOL. We then utilize these formal definitions to verify the expectation and variance characteristics of the Geometric random variable. In order to demonstrate the practical effectiveness of the formalization presented in this paper, we also present the probabilistic analysis of the Coupon Collector's problem in HOL.

Asymptotic approximations for coupon collectors

A collector samples with replacement a set of n ≧ 2 distinct coupons until he has n − m , 0 ≦ m < n , distinct coupons for the first time. We refine the limit theorems concerning the standardized random number of necessary draws if n → ∞ and m is fixed: we give a one-term asymptotic expansion of the distribution function in question, providing a better approximation of it, than the one given by the limiting distribution function, and proving in particular that the rate of convergence in these limiting theorems is of order (log n )/ n .

The Generalised Coupon Collector Problem

Coupons are collected one at a time from a population containingndistinct types of coupon. The process is repeated until allncoupons have been collected and the total number of draws,Y, from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distributionXwith the probability that a type-icoupon is drawn being P(X=i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distributionY(appropriately normalized) as the number of couponsn→ ∞ under general assumptions upon the asymptotic distribution ofX. The results are proved by studying the total number of coupons,W(t), not collected intdraws from the population and noting that P(Y≤t) = P(W(t) = 0). Two normalizations ofYare considered, the choice of normalization depending upon whether or not a suitable Poisson limit exists forW(t). Finally, extensions to theK-coupon collector problem and the birthday problem are given.

The Generalised Coupon Collector Problem

Coupons are collected one at a time from a population containing n distinct types of coupon. The process is repeated until all n coupons have been collected and the total number of draws, Y, from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distribution X with the probability that a type-i coupon is drawn being P(X = i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distribution Y (appropriately normalized) as the number of coupons n → ∞ under general assumptions upon the asymptotic distribution of X. The results are proved by studying the total number of coupons, W(t), not collected in t draws from the population and noting that P(Y ≤ t) = P(W(t) = 0). Two normalizations of Y are considered, the choice of normalization depending upon whether or not a suitable Poisson limit exists for W(t). Finally, extensions to the K-coupon collector problem and the birthday problem are given.

Coupon Collecting with Quotas

We analyze a variant of the coupon collector's problem, in which the probabilities of obtaining coupons and the numbers of coupons in a collection may be non-uniform. We obtain a finite expression for the generating function of the probabilities to complete a collection and show how this generalizes several previous results about the coupon collector's problem. Also, we provide applications about computational complexity and approximation.

Formal verification of tail distribution bounds in the HOL theorem prover

AbstractTail distribution bounds play a major role in the estimation of failure probabilities in performance and reliability analysis of systems. They are usually estimated using Markov's and Chebyshev's inequalities, which represent tail distribution bounds for a random variable in terms of its mean or variance. This paper presents the formal verification of Markov's and Chebyshev's inequalities for discrete random variables using a higher‐order‐logic theorem prover. The paper also provides the formal verification of mean and variance relations for some of the widely used discrete random variables, such as Uniform(m), Bernoulli(p), Geometric(p) and Binomial(m, p) random variables. This infrastructure allows us to precisely reason about the tail distribution properties and thus turns out to be quite useful for the analysis of systems used in safety‐critical domains, such as space, medicine or transportation. For illustration purposes, we present the performance analysis of the coupon collector's problem, a well‐known commercially used algorithm. Copyright © 2008 John Wiley & Sons, Ltd.

Methods for Studying Generalized Birthday and Coupon Collection Problems

In this paper, we consider generalizations of two classical probability problems: the birthday problem and the coupon collector's problem. These problems are discussed in terms of urn models and captured through generating functions. Some methods for the study of the problems are presented. Furthermore, we also formulate the generalized birthday and coupon collector's problems as the waiting time problems. In each case, numerical examples are given in order to illustrate the feasibility of our methods.

COUPON COLLECTING

We consider the classical coupon collector's problem in which each new coupon collected is type i with probability pi; ∑i=1npi=1. We derive some formulas concerning N, the number of coupons needed to have a complete set of at least one of each type, that are computationally useful when n is not too large. We also present efficient simulation procedures for determining P(N > k), as well as analytic bounds for this probability.

A Survey of the Coupon Collector’s Problem with Random Sample Sizes

This paper surveys the coupon collector’s waiting time problem with random sample sizes and equally likely balls. Consider an urn containing m red balls. For each draw, a random number of balls are removed from the urn. The group of removed balls is painted white and returned to the urn. Several approaches to addressing this problem are discussed, including a Markov chain approach to compute the distribution and expected value of the number of draws required for the urn to contain j white balls given that it currently contains i white balls. As a special case, E[N], the expected number of draws until all the balls are white given that all are currently red is also obtained.

Biased random-to-top shuffling

Recently Wilson [Ann. Appl. Probab. 14 (2004) 274–325] introduced an important new technique for lower bounding the mixing time of a Markov chain. In this paper we extend Wilson’s technique to find lower bounds of the correct order for card shuffling Markov chains where at each time step a random card is picked and put at the top of the deck. Two classes of such shuffles are addressed, one where the probability that a given card is picked at a given time step depends on its identity, the so-called move-to-front scheme, and one where it depends on its position. For the move-to-front scheme, a test function that is a combination of several different eigenvectors of the transition matrix is used. A general method for finding and using such a test function, under a natural negative dependence condition, is introduced. It is shown that the correct order of the mixing time is given by the biased coupon collector’s problem corresponding to the move-to-front scheme at hand. For the second class, a version of Wilson’s technique for complex-valued eigenvalues/eigenvectors is used. Such variants were presented in [Random Walks and Geometry (2004) 515–532] and [Electron. Comm. Probab. 8 (2003) 77–85]. Here we present another such variant which seems to be the most natural one for this particular class of problems. To find the eigenvalues for the general case of the second class of problems is difficult, so we restrict attention to two special cases. In the first case the card that is moved to the top is picked uniformly at random from the bottom k=k(n)=o(n) cards, and we find the lower bound (n3/(4π2k(k−1)))logn. Via a coupling, an upper bound exceeding this by only a factor 4 is found. This generalizes Wilson’s [Electron. Comm. Probab. 8 (2003) 77–85] result on the Rudvalis shuffle and Goel’s [Ann. Appl. Probab. 16 (2006) 30–55] result on top-to-bottom shuffles. In the second case the card moved to the top is, with probability 1/2, the bottom card and with probability 1/2, the card at position n−k. Here the lower bound is again of order (n3/k2)logn, but in this case this does not seem to be tight unless k=O(1). What the correct order of mixing is in this case is an open question. We show that when k=n/2, it is at least Θ(n2).

Some upper and lower bounds on the coupon collector problem

The classical coupon collector problem is closely related to probabilistic-packet-marking (PPM) schemes for IP traceback problem in the Internet. In this paper, we study the classical coupon collector problem, and derive some upper and lower bounds of the complementary cumulative distribution function (ccdf) of the number of objects (coupons) that one has to check in order to detect a set of different objects. The derived bounds require much less computation than the exact formula. We numerically find that the proposed bounds are very close to the actual ccdf when detecting probabilities are set to the values common to the PPM schemes.

Some New Aspects of the Coupon Collector's Problem

We extend the classical coupon collector's problem to one in which two collectors are simultaneously and independently seeking collections of d coupons. We find, in finite terms, the probability that the two collectors finish at the same trial, and we find, using the methods of Gessel and Viennot, the probability that the game has the following "ballot-like" character: the two collectors are tied with each other for some initial number of steps, and after that the player who first gains the lead remains ahead throughout the game. As a by-product we obtain the evaluation in finite terms of certain infinite series whose coefficients are powers and products of Stirling numbers of the second kind. We study the variant of the original coupon collector's problem in which a single collector wants to obtain at least h copies of each coupon. Here we give a simpler derivation of results of Newman and Shepp and extend those results. Finally, we obtain the distribution of the number of coupons that have been obtained exactly once ("singletons") at the conclusion of a successful coupon collecting sequence.

A coupon collector's problem with bonuses

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.

Some exactly solvable models of urn process theory

We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over curves of the Fermat type (which are also hypergeometric functions), together with their inverses. A consequence is the unification of the analyses of many classical models, including those related to the coupon collector's problem, particle transfer (the Ehrenfest model), Friedman's "adverse campaign'' and Pólya's contagion model, as well as the OK Corral model (a basic case of Lanchester's theory of conflicts). In each case, it is possible to quantify very precisely the probable composition of the urn at any discrete instant. We study here in detail "semi-sacrificial'' urns, for which the following are obtained: a Gaussian limiting distribution with speed of convergence estimates as well as a characterization of the large and extreme large deviation regimes. We also work out explicitly the case of $2$-dimensional triangular models, where local limit laws of the stable type are obtained. A few models of dimension three or greater, e.g., "autistic'' (generalized Pólya), cyclic chambers (generalized Ehrenfest), generalized coupon-collector, and triangular urns, are also shown to be exactly solvable.

Martingale approach to the coupon collection problem

A new method for obtaining different moment characteristics in the coupon collector’s problem is developed. Various martingale corollaries are obtained. Bibliography: 9 titles.

Exchangeable pairs and Poisson approximation

This is a survery paper on Poisson approximation using Stein's method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector's problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered.

Large deviation asymptotics for occupancy problems

In the standard formulation of the occupancy problem one considers the distribution of r balls in n cells, with each ball assigned independently to a given cell with probability 1/n. Although closed form expressions can be given for the distribution of various interesting quantities (such as the fraction of cells that contain a given number of balls), these expressions are often of limited practical use. Approximations provide an attractive alternative, and in the present paper we consider a large deviation approximation as r and n tend to infinity. In order to analyze the problem we first consider a dynamical model, where the balls are placed in the cells sequentially and “time” corresponds to the number of balls that have already been thrown. A complete large deviation analysis of this “process level” problem is carried out, and the rate function for the original problem is then obtained via the contraction principle. The variational problem that characterizes this rate function is analyzed, and a fairly complete and explicit solution is obtained. The minimizing trajectories and minimal cost are identified up to two constants, and the constants are characterized as the unique solution to an elementary fixed point problem. These results are then used to solve a number of interesting problems, including an overflow problem and the partial coupon collector’s problem.

Some bounds on the coupon collector problem

AbstractThis work addresses on the coupon collector problem and its generalization introduced by Flajolet, Gardy, and Thimonier. In our main results, we show a ratio limit theorem for the random time of the generalized coupon collector problem, and, further, we give the leading term and the geometric rate for the distribution of this random time, when the number of throws is large. For the classical coupon collector problem, we give a bound on the conditional second moment for the number of visits to the coupons, relying strongly on a result of Holst on extremal distributions. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004