scholarly journals Optimization results for a generalized coupon collector problem

2016 ◽  
Vol 53 (2) ◽  
pp. 622-629 ◽  
Author(s):  
Emmanuelle Anceaume ◽  
Yann Busnel ◽  
Ernst Schulte-Geers ◽  
Bruno Sericola
Keyword(s):  
Probability Distribution ◽  
Uniform Distribution ◽  
Closed Subset ◽  
Probability Distributions ◽  
Fixed Number ◽  
Coupon Collector ◽  
The One

Abstract In this paper we study a generalized coupon collector problem, which consists of analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one which stochastically maximizes the time needed to collect a fixed number of distinct coupons.

scholarly journals New Results on a Generalized Coupon Collector Problem Using Markov Chains

2015 ◽  
Vol 52 (02) ◽  
pp. 405-418 ◽  
Author(s):  
Emmanuelle Anceaume ◽  
Yann Busnel ◽  
Bruno Sericola
Keyword(s):  
Probability Distribution ◽  
Markov Chains ◽  
Uniform Distribution ◽  
Fixed Number ◽  
Expected Time ◽  
Complementary Distribution ◽  
Optimization Result ◽  
Fixed Subset ◽  
Coupon Collector ◽  
The Way

In this paper we study a generalized coupon collector problem, which consists of determining the distribution and the moments of the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we obtain expressions for the distribution and the moments of this time. We also prove that the almost-uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which minimizes the expected time to obtain a fixed subset of distinct coupons. This optimization result is extended to the complementary distribution of the time needed to obtain the full collection, proving by the way this well-known conjecture. Finally, we propose a new conjecture which expresses the fact that the almost-uniform distribution should minimize the complementary distribution of the time needed to obtain any fixed number of distinct coupons.

scholarly journals New Results on a Generalized Coupon Collector Problem Using Markov Chains

2015 ◽  
Vol 52 (2) ◽  
pp. 405-418 ◽  
Author(s):  
Emmanuelle Anceaume ◽  
Yann Busnel ◽  
Bruno Sericola
Keyword(s):  
Probability Distribution ◽  
Markov Chains ◽  
Uniform Distribution ◽  
Fixed Number ◽  
Expected Time ◽  
Complementary Distribution ◽  
Optimization Result ◽  
Fixed Subset ◽  
Coupon Collector ◽  
The Way

In this paper we study a generalized coupon collector problem, which consists of determining the distribution and the moments of the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we obtain expressions for the distribution and the moments of this time. We also prove that the almost-uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which minimizes the expected time to obtain a fixed subset of distinct coupons. This optimization result is extended to the complementary distribution of the time needed to obtain the full collection, proving by the way this well-known conjecture. Finally, we propose a new conjecture which expresses the fact that the almost-uniform distribution should minimize the complementary distribution of the time needed to obtain any fixed number of distinct coupons.

scholarly journals Diachronic changes in word probability distributions in daily press

Psihologija ◽  
2006 ◽  
Vol 39 (1) ◽  
pp. 75-89
Author(s):  
Jelena Stankovic ◽  
Aleksandar Kostic
Keyword(s):  
Probability Distribution ◽  
Probability Distributions ◽  
Sentence Length ◽  
The Other ◽  
Daily News ◽  
Daily Press ◽  
Two Samples ◽  
The One ◽  
Diachronic Changes

Changes in probability distributions of individual words and word types were investigated within two samples of daily press in the span of fifty years. Two samples of daily press were used in this study. The one derived from the Corpus of Serbian Language (CSL) /Kostic, Dj., 2001/ that covers period between 1945. and 1957. and the other derived from the Ebart Media Documentation (EBR) that was complied from seven daily news and five weekly magazines from 2002. and 2003. Each sample consisted of about 1 million words. The obtained results indicate that nouns and adjectives were more frequent in the CSL, while verbs and prepositions are more frequent in the EBR sample, suggesting a decrease of sentence length in the last five decades. Conspicuous changes in probability distribution of individual words were observed for nouns and adjectives, while minimal or no changes were observed for verbs and prepositions. Such an outcome suggests that nouns and adjectives are most susceptible to diachronic changes, while verbs and prepositions appear to be resistant to such changes.

scholarly journals Exponential Negation of a Probability Distribution

2021 ◽  
Author(s):  
Qinyuan Wu ◽  
Yong Deng ◽  
Neal Xiong
Keyword(s):  
Information Processing ◽  
Probability Distribution ◽  
Uniform Distribution ◽  
Probability Distributions ◽  
Numerical Examples ◽  
Entropy Increase ◽  
Intelligent Information Processing ◽  
Uniform Probability Distribution ◽  
Intelligent Information ◽  
Number Of Iterations

Abstract Negation operation is important in intelligent information processing. Different with existing arithmetic negation, an exponential negation is presented in this paper. The new negation can be seen as a kind of geometry negation. Some basic properties of the proposed negation are investigated, we find that the fix point is the uniform probability distribution. The proposed exponential negation is an entropy increase operation and all the probability distributions will converge to the uniform distribution after multiple negation iterations. The convergence speed of the proposed negation is also faster than the existed negation. The number of iterations of convergence is inversely proportional to the number of elements in the distribution. Some numerical examples are used to illustrate the efficiency of the proposed negation.

scholarly journals Rainfall models – a study over Gangtok

MAUSAM ◽  
2021 ◽  
Vol 61 (2) ◽  
pp. 225-228
Author(s):  
K. SEETHARAM
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Uniform Distribution ◽  
Probability Distribution Function ◽  
Null Hypothesis ◽  
Probability Distributions ◽  
Data Sets ◽  
Type I ◽  
Anderson Darling ◽  
Pearsonian Type

In this paper, the Pearsonian system of curves were fitted to the monthly rainfalls from January to December, in addition to the seasonal as well as annual rainfalls totalling to 14 data sets of the period 1957-2005 with 49 years of duration for the station Gangtok to determine the probability distribution function of these data sets. The study indicated that the monthly rainfall of July and summer monsoon seasonal rainfall did not fit in to any of the Pearsonian system of curves, but the monthly rainfalls of other months and the annual rainfalls of Gangtok station indicated to fit into Pearsonian type-I distribution which in other words is an uniform distribution. Anderson-Darling test was applied to for null hypothesis. The test indicated the acceptance of null-hypothesis. The statistics of the data sets and their probability distributions are discussed in this paper.

scholarly journals Probability estimate and the optimal text size

Psihologija ◽  
2008 ◽  
Vol 41 (1) ◽  
pp. 35-51 ◽  
Author(s):  
Aleksandar Kostic ◽  
Svetlana Ilic ◽  
Petar Milin
Keyword(s):  
Probability Distribution ◽  
Sample Size ◽  
Probability Distributions ◽  
A Priori ◽  
Optimal Sample Size ◽  
Minimal Sample ◽  
Optimal Sample ◽  
Geometrical Progression ◽  
The One ◽  
Stable Probability Distributions

Reliable language corpus implies a text sample of size n that provides stable probability distributions of linguistic phenomena. The question is what is the minimal (i.e. the optimal) text size at which probabilities of linguistic phenomena become stable. Specifically, we were interested in probabilities of grammatical forms. We started with an a priori assumption that text size of 1.000.000 words is sufficient to provide stable probability distributions. Text of this size we treated as a "quasi-population". Probability distribution derived from the "quasi-population" was then correlated with probability distribution obtained on a minimal sample size (32 items) for a given linguistic category (e.g. nouns). Correlation coefficient was treated as a measure of similarity between the two probability distributions. The minimal sample was increased by geometrical progression, up to the size where correlation between distribution derived from the quasi-population and the one derived from an increased sample reached its maximum (r=1). Optimal sample size was established for grammatical forms of nouns, adjectives and verbs. General formalism is proposed that allows estimate of an optimal sample size from minimal sample (i.e. 32 items).

Interpretation of Results of Penetration Tests Performed on Timber Structures in Bending

2013 ◽  
Vol 486 ◽  
pp. 347-352 ◽  
Author(s):  
Lenka Melzerová ◽  
Michal Šejnoha
Keyword(s):  
Probability Distribution ◽  
Measurement Errors ◽  
Elastic Moduli ◽  
Probability Distributions ◽  
Load Level ◽  
Statistical Dependence ◽  
Fiber Direction ◽  
The One ◽  
Penetration Tests

The paper concentrates on the determination of local elastic moduli of timber in the fiber direction. To that end a single commercially produced glued timber beam was subjected to 3600 penetration measurements. The beam was first covered by a regular grid of monitoring points at which the depth of indentation was measured. The pin was shot into the wood with a given energy (Pilodin 6J). At the same time the moisture content at 80 selected points was determined. A specific measurement order was assumed to exclude principal measurement errors on the one hand and to minimize systematic measurement errors on the other. We expect the measured elastic moduli to serve as an input for advanced finite element simulations on the bases of stochastic analysis. In such a case the local measured moduli represent in a given segment of each lamella an ensemble of data characterized by a selected probability distribution. These distributions are then employed in the LHS based stochastic simulation to provide probability distribution of the maximum deflection for a given load level. Apart from that it appears meaningful to compare independently the probability distributions of the elastic moduli for segments of the lamella (these may considerably differ owing to the specifics of the production of structures made from glued lamella timber) with statistical data from the whole beam. Based on the measured data the correlation matrix relating statistical dependence of individual segments can be estimated thus improving the quality of the stochastic model.

scholarly journals Rate of convergence for traditional Pólya urns

2020 ◽  
Vol 57 (4) ◽  
pp. 1029-1044
Author(s):  
Svante Janson
Keyword(s):  
Rate Of Convergence ◽  
Dirichlet Distribution ◽  
Fixed Number ◽  
Kolmogorov Distance ◽  
Initial Composition ◽  
Exact Distributions ◽  
Polya Urn ◽  
The One ◽  
Lévy Distance ◽  
Direct Calculations

AbstractConsider a Pólya urn with balls of several colours, where balls are drawn sequentially and each drawn ball is immediately replaced together with a fixed number of balls of the same colour. It is well known that the proportions of balls of the different colours converge in distribution to a Dirichlet distribution. We show that the rate of convergence is $\Theta(1/n)$ in the minimal $L_p$ metric for any $p\in[1,\infty]$, extending a result by Goldstein and Reinert; we further show the same rate for the Lévy distance, while the rate for the Kolmogorov distance depends on the parameters, i.e. on the initial composition of the urn. The method used here differs from the one used by Goldstein and Reinert, and uses direct calculations based on the known exact distributions.

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

WIGNER FUNCTION OF PULSED FIELDS RECONSTRUCTED BY DIRECT DETECTION

2011 ◽  
Vol 09 (supp01) ◽  
pp. 39-47
Author(s):  
ALESSIA ALLEVI ◽  
MARIA BONDANI ◽  
ALESSANDRA ANDREONI
Keyword(s):  
Probability Distribution ◽  
Wigner Function ◽  
Beam Splitter ◽  
Direct Detection ◽  
Probability Distributions ◽  
Linear Regime ◽  
Wigner Functions ◽  
Intensity Measurements ◽  
Pulsed Fields

We present the experimental reconstruction of the Wigner function of some optical states. The method is based on direct intensity measurements by non-ideal photodetectors operated in the linear regime. The signal state is mixed at a beam-splitter with a set of coherent probes of known complex amplitudes and the probability distribution of the detected photons is measured. The Wigner function is given by a suitable sum of these probability distributions measured for different values of the probe. For comparison, the same data are analyzed to obtain the number distributions and the Wigner functions for photons.

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