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Scenario programs have established themselves as efficient tools toward decision-making under uncertainty. To assess the quality of scenario-based solutions a posteriori , validation tests based on Bernoulli trials have been widely adopted in practice. However, to reach a theoretically reliable judgment of risk, one typically needs to collect massive validation samples. In this article, we propose new a posteriori bounds for convex scenario programs with validation tests, which are dependent on both realizations of support constraints, and performance on out-of-sample validation data. The proposed bounds enjoy wide generality in that many existing theoretical results can be incorporated as particular cases. To facilitate practical use, a systematic approach for parameterizing a posteriori probability bounds is also developed, which is shown to possess a variety of desirable properties allowing for easy implementations and clear interpretations. By synthesizing comprehensive information about support constraints and validation tests, improved risk evaluation can be achieved for randomized solutions in comparison with existing a posteriori bounds. Case studies on controller design of aircraft lateral motion are presented to validate the effectiveness of the proposed a posteriori bounds.

Conventional sampling in biostatistics and economics posits an individual in a fixed observable state (e.g., diseased or not, poor or not, etc.). Social, market, and opinion research, however, require a cognitive sampling theory which recognizes that a respondent has a choice between two options (e.g., yes versus no). This new theory posits the survey re- spondent as a personal probability. Once the sample is drawn, a series of independent non-identical Bernoulli trials are carried out. The outcome of each trial is a momentary binary choice governed by this unobserved proba- bility. Liapunov's extended central limit theorem (Lehmann, 1999) and the Horvitz-Thompson (1952) theorem are then brought to bear on sampling unobservables, in contrast to sampling observations. This formulation reaf- firms the usefulness of a weighted sample proportion, which is now seen to estimate a different target parameter than that of conventional design-based sampling theory.

As far as the agricultural domain is concerned, one of the most hot research areas of analysis is accurate prediction of leaf disease from the leaf images of a plant. The prediction of agricultural plant diseases bymeans of the image processing techniques will hence reduce the dependence on the farmers to safeguard their agricultural land and also their products. However, with the presence of noise, the leaf disease prediction is said to be hindered. To address this issue, in this paper, Covariance Kalman Geometric Graph-basedBernoulliClassifier (CKGG-BC) for Plant leaf disease prediction is proposed. The CKGG-BC method is split into three parts. To start with the plant leaf image provided as input, the Covariance Kalman Filtered Preprocessing modelintroduced for the image enhancement. Second, Geometric Graph-based Segmented Co-occurrence Feature Extraction model is applied to the preprocessed image to accurately segment the infected leaf areas and followed by which extracting the accurate infected leaf areas. Finally, Bernoulli Online Multiple Kernel Learning Classifier is applied for accurate plant leaf disease prediction with minimum classification error. The proposed method provides a significant refinement with respect to state-of-the-art methods. Even under complex background conditions, i.e., in the presence of noise, the averageaccuracy of the proposed method is said to be improved and hence paves mechanism for prediction of plant leaf disease in a significant manner. Experimentalresults exhibit the effectiveness of the proposed method in terms of computational overhead, accuracy, true positive rate and classification error respectively.

The use of virtual species to test species distribution models is important for understanding how aspects of the model development process influence model performance. Typically, virtual species are simulated by defining their niche as a function of environmental variables and simulating occurrence probabilistically via Bernoulli trials. This approach ignores endogenous processes known to drive species distribution such as dispersal and population dynamics. To understand whether these processes are important for simulating virtual species we compared the probabilistic simulation approach to those incorporating endogenous processes. This comparison was done by evaluating changes in the relationship between species occurrence and habitat suitability over a number of landscapes with varying spatial structure. We found that the combined effects of population dynamics and dispersal meant the probability of occurrence of a single cell was not only dependent on habitat suitability, but also the number of occupied cells nearby. This resulted in a dependence on the size of clusters of high suitability cells (analogous to patch size) to maintain populations, increased residual spatial autocorrelation and non‐stationarity of the species response between landscapes. These data characteristics are attributes of real species distribution data and are not present in probabilistic simulations. Researchers using virtual species should consider the importance of these characteristics to their study objectives to decide whether the inclusion of endogenous processes is necessary.

This paper investigates the queue-dependent service rates(QDSR) in the stochastic queueing-inventory system(SQIS). This SQIS consists a single server service channel, $ S $ number of inventories, and a finite queue. An arriving customer gets the service immediately if the server is free and there exists a positive stock in the SQIS. When the server is busy, they have to wait in the finite queue. Suppose they find that the waiting hall is full, either they leave the system or enter into an infinite orbit under the Bernoulli trial. The service rate of any arrival is dependent on the number of customers in the queue at present. The orbital customer can compete for the service only through joining into the waiting hall based on the classical retrial policy. Whenever the number of inventories in SQIS is reached a certain limit $ s $, the replenishment of $ Q( = S-s) $ items is placed. Due to the structure of rate matrix, the stability analysis, minimal non-negative solutions of the quadratic systems are derived through the Neuts matrix-geometric approximation(MGA). Further, the waiting time distribution(WTD) of arrival and necessary system characteristics are derived. Finally, adequate numerical examples are presented to highlight the proposed SQIS.

The chemical sensing process can be understood as the physical process consisting of binding and unbinding reactions on the sensing surface. The random nature of the reactions lead to fluctuations in the response of a sensor. The response of the sensor is modeled in the form of a stochastic differential equation (SDE). The model is developed by formulating the binding and unbinding reactions in the form of Bernoulli trials. The transition probability density function (pdf) of the number of binded analyte molecules $X_t$ , at a given time instant $t$ , has been evaluated using the Bernoulli trial representation. The obtained pdf is used to derive the Fokker Planck equation (FPE) for the stochastic process. The FPE has a direct correspondence with the desired SDE model. A maximum aposteriori (MAP) estimation method has been proposed for evaluating the parameters. The SDE model for the fluctuations in the sensor signal has been evaluated. The frequency spectrum of the fluctuations is evaluated using the SDE. The proposed model is used to analyze noise due to the ligand-receptor kinetics in the receiver in molecular communication. The utility of the model in agent identification applications has been demonstrated.

The use of statistical modeling for the development of information and mathematical culture of schoolchildren is considered in the article. A comparative analysis of various statistical modeling tools: spreadsheets, programming languages, interactive mathematical environments is carried out. Examples of the simplest statistical models and their implementation using these tools are given: experiments with coins and cubes, random selection without returning, Bernoulli tests before the first success. The most interesting theoretical issues that arise during statistical experiments and are available for study in the school mathematics course are highlighted. Several games with probabilistic content are considered, demonstrating well-known statistical paradoxes. It is shown how the skills accumulated in the process of statistical modeling can be used to study real situations and perform independent research and design work on this basis.

The goodness of fit of first motion manually picked polarities from the reviewed Bulletin of the International Seismological Centre (ISC) is examined versus available earthquake mechanism solutions in order to study the behavior of seismic stations, and/or reporting agencies, over time. The probability of the reporting polarity data fitting (and not fitting) source mechanisms is quantified by calculating the probability distribution of several Bernoulli trials over a randomly perturbed set of hypocenters and velocity models for each earthquake mechanism–station polarity combination. The overall agreement of first motion polarities against source mechanisms is found to be good with a few cases of seismic stations showing indications of systematic phase reversals over certain time periods. Specifically, results were obtained for 50% of the registered stations at the ISC, and from these stations 70% show reliable operation during the operational time period under investigation, with only 3% showing the opposite, and 7% showing evidence of systematic changes in the quality of the reported first motion polarities. The rest showed great variability over short periods of time, which does not allow one to draw any conclusions. Comparing waveform data with the associated reported polarities revealed a mixture of cases of either questionable picking or true station phase reversals. The ISC is planning to make available on its website the detailed results for each station in order to make data users and seismic network operators aware of cases that need attention.

Probability and expectation are two distinct measures, both of which can be used to indicate the likelihood of certain events. However, expectation values, which are often associated with waiting times for success, may, at times, speak more clearly and poignantly about the uncertainty of an event than a theoretical probability. To illustrate the point, suppose the probability of choosing a winning lottery ticket is 2.5 × 10−8. This information may not communicate the unlikely odds of winning as clearly as a statement like, “If five lottery tickets are purchased per day, the expected waiting time for a first win is about 22000 years.”

Data Centers are the backbone of any industry that provides a specialized environment to safeguard the company's valuable equipment and intellectual property. The successful operation of company's Data Center is typically shared among multiple departments and personnel. But, the involvement of multiple personal and departments make its configuration complex. In such situations, reliability of data center become a necessary factor. But, the traditional theory of reliability is based on the Bernoulli trials, i.e., either operative or failure. But this situation is unrealistic in case of complex systems like data center. To rectify this problem, here a mathematical model has been developed using the concept of fuzzy reliability. All the failure and repair rates are exponential distributed along with coverage factor. Chapman-Kolmogorov differential equations have been developed for the fuzzy system using Markov birth–death process. A new methodology Runge–Kutta method of order four has been used to solve Chapman-Kolmogorov differential equations using MATLAB (Ode 45 function).

We show that for good measures, the set of homeomorphisms of Cantor space which preserve that measure and which have no invariant clopen sets contains a residual set of homeomorphisms which are uniquely ergodic. Additionally, we show that for refinable Bernoulli trial measures, the same set of homeomorphisms contains a residual set of homeomorphisms which admit only finitely many ergodic measures.

We present a method of generating functions to compute the distributions of the first-arrival and inter-arrival times of random patterns in independent Bernoulli trials and first-order Markov trial...

The main difficulties when using the Bayesian approach are obtaining information from the specialist and obtaining hyperparameters values of the assumed probability distribution as representative of knowledge external to the data. In addition to the fact that a large part of the literature on this subject is characterized by considering prior conjugated distributions for the parameter of interest. An method is proposed to find the hyperparameters of a nonconjugated prior distribution. The following scenarios were considered for Bernoulli trials: four prior distributions (Beta, Kumaraswamy, Truncated Gamma and Truncated Weibull) and four scenarios for the generating process. Two necessary, but not sufficient conditions were identified to ensure the existence of a vector of values for the hyperparameter. The Truncated Weibull prior distribution performed the worst. The methodology was used to estimate the prevalence of two transmitted sexually infections in an Colombian indigenous community.

The notion of a Bernoulli trial is extended, by introducing recursively different kinds (ranks, levels) of success and failure, to a Bernoulli trial with chain-composite success (or failure). Then,...

This paper presents a probabilistic model of cumulative damage based on Markov chains theory to model propagation of internal corrosion depth localized in a hydrocarbons transport pipeline. The damage accumulation mechanism is unit jump type, depending on the state. It uses a shock model based on Bernoulli trials and probabilities to remain in the same state or the next one. Data are adjusted to Lognormal distribution and proven with a Kolmogorov-Smirnov test. The vector obtained from multiplying the initial state vector with the transition matrix was developed and the system of equations to find each transition probability with a single inspection report was solved. In order to calculate propagation of internal corrosion after inspection, an exponential equation was proposed and a parameter was adjusted to the data. Time to expected failure was obtained by adding the time expected in each damage state. Each time step was adjusted to real time.

We examine the concept of probability from its emergence within the realm of the games of chance and the development of the theory of probability until the appearance of the treatise of Kolmogorov on this subject. The discipline related to that theory is framed as the science of aleatory events. Probability is understood as a primitive concept represented by a measure assigned to the space of events and obeying the fundamental postulates of the theory. The measurement of probability is the ratio of the number of the observed favorable outcomes and the total number of observed outcomes when these numbers are larger enough.

Medical studies often involve a comparison between two outcomes, each collected from a sample. The probability associated with, and confidence in the result of the study is of most importance, since one may argue that having been wrong with a percent could be what killed a patient. Sampling is usually done from a finite and discrete population and it follows a Bernoulli trial, leading to a contingency of two binomially distributed samples (better known as 2×2 contingency table). Current guidelines recommend reporting relative measures of association (such as the relative risk and odds ratio) in conjunction with absolute measures of association (which include risk difference or excess risk). Because the distribution is discrete, the evaluation of the exact confidence interval for either of those measures of association is a mathematical challenge. Some alternate scenarios were analyzed (continuous vs. discrete; hypergeometric vs. binomial), and in the main case—bivariate binomial experiment—a strategy for providing exact p-values and confidence intervals is proposed. Algorithms implementing the strategy are given.

Many ecological applications, like the study of mortality rates, require the estimation of proportions and confidence intervals for them. The traditional way of doing this applies the binomial distribution, which describes the outcome of a series of Bernoulli trials. This distribution assumes that observations are independent and the probability of success is the same for all the individual observations. Both assumptions are obviously false in many cases. I show how to apply bootstrap and the Poisson binomial distribution (a generalization of the binomial distribution) to the estimation of proportions. Any information at the individual level would result in better (narrower) confidence intervals around the estimation of proportions. As a case study, I applied this method to the calculation of mortality rates in a forest plot of tropical trees in Lambir Hills National Park, Malaysia. I calculated central estimates and 95% confidence intervals for species‐level mortality rates for 1,007 tree species. I used a very simple model of spatial dependence in survival to estimate individual‐level risk of mortality. The results obtained by accounting for heterogeneity in individual‐level risk of mortality were comparable to those obtained with the binomial distribution in terms of central estimates, but the precision increased in virtually all cases, with an average reduction in the width of the confidence interval of ~20%. Spatial information allows the estimation of individual‐level probabilities of survival, and this increases the precision in the estimates of mortality rates. The general method described here, with modifications, could be applied to reduce uncertainty in the estimation of proportions related to any spatially structured phenomenon with two possible outcomes. More sophisticated approaches can yield better estimates of individual‐level mortality and thus narrower confidence intervals.

In this paper, we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions, which are mixed Poisson distributions with the mixing laws belonging to the class of generalized gamma (GG) distributions. This family was introduced by E.W. Stacy as a particular family of lifetime distributions containing gamma, exponential power, and Weibull distributions. These distributions seem to be very promising in the statistical description of many real phenomena and very convenient and almost universal models for the description of statistical regularities in discrete data. We study analytic properties of GNB distributions. We prove that a GG distribution is a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. We write the mixing distribution explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative half-line. As a corollary, we obtain a representation for the GNB distribution as a mixed geometric distribution. We consider the corresponding scheme of Bernoulli trials with random probability of success. Within this scheme, we prove a random analog of the Poisson theorem establishing the convergence of mixed binomial distributions to mixed Poisson laws. We prove limit theorems for random sums of independent random variables in which the number of summands has the GNB distribution and the summands have both light- and heavy-tailed distributions. We obtain that the class of limit laws is wide enough and includes the so-called generalized variance gamma distributions. We obtain various representations for the limit laws in terms of mixtures of Mittag-Leffler, Linnik, or Laplace distributions. We prove limit theorems establishing the convergence of the distributions of statistics constructed from samples with random sizes obeying the GNB distribution to generalized variance gamma distributions. We also discuss some applications of GNB distributions in meteorology.

Recently Khmaladze has shown how to ‘rotate’ one empirical process to another. We apply this methodology to goodness of fit tests for Bernoulli trials, generated by a single distributional family, but with covariates varying over the sample. Grouping the data, we demonstrate that goodness of fit tests after rotation to distribution free processes are easily computed, and exhibit high power to reject incorrect null hypotheses.

In the present article, we study the classic Bernoulli weak law of large numbers and Borel strong law of large numbers, which weaken the assumptions of some known results.

This paper presents a new and widely applicable nonparametric approach to the characterisation of time series dynamics. The approach involves analysis of the incidence of occurrence of patterns in the direction of movement of the series, and may readily be applied to time series data measured on any scale. The paper includes derivations of analytic forms for two (infinite) families of distributions under the null hypothesis of random behaviour, and of a useful analytic form for the generation of the moments of these distributions. The distributions are asymptotically normal, so allowing for straightforward application of the approach presented in the paper too long series of high frequency and/or extended time period data. Areas of application in finance and accounting are suggested.

A reputation system is proposed in the form of a Reputation Network based on the premise of availability of a model of the world as a Dynamic Bayesian Network where the network's posterior for one time slice is set as the prior probability in the next time slice. These networks are solutions with better overall properties than the original network reducing the complexity of using the results when the input data is random. Experiments are an extension of prior work detecting when P2P Group Owner (GO) negotiation unfairly makes one of the participants GO more often than the other. The decision process of who becomes GO can be modeled as a Bernoulli trial. Based on this model, we show that we can set the false positive rate, we can detect more nuanced cheating without extra nodes in the network to help us, and when the underlying distribution favors one element as GO, it will declare a positive and keep the positive set without needing a second interpretive layer. Furthermore, we demonstrate that the original properties of non-random data are maintained.

Micro-electromechanical systems (MEMS) have developed rapidly in recent years in various technical fields that have increased their interest in the Direct Simulation Monte Carlo (DSMC) method. In this paper, we present a simple representation of the DSMC collision scheme and investigate the correctness of the Simplified Bernoulli Trial (SBT) collision scheme for the calculation of two-dimensional flows. The first part of the collision scheme, which determines collision pairs, is presented following the derivation of the expression for the mean free path and using the cumulative distribution function. Approaches and conclusions based on one-dimensional flows are not always directly applicable to two- and three-dimensional flows. We investigated SBT correctness by using the two-dimensional pressure-driven gas flow of monoatomic gas as a test case. We studied the influence of shuffling of the list of particles per cell (PPC) before the collision scheme&rsquo, s execution, as well as the minimal and maximal number of PPC, on the correctness of the solution. The investigation showed that shuffling and the number of PPC played an important role in the correctness of SBT. Our recommendations are straightforwardly applicable to three-dimensional flows. Finally, we considered the mixing of two gases and compared the results available in the literature.

In the branch of forensic science known as firearm evidence identification, various similarity scores have been proposed to compare firearm marks. Some similarity score comparisons, for example, congruent matching cells (CMC) method, are based on pass-or-fail tests. The CMC method compares the pairwise topography images of breech face impressions, from which the similarity score is derived for quantifying their topography similarity. For an image pair, the CMC method determines a certain number of correlated cell pairs. Next, each correlated pair is determined to be a congruent match cell (CMC) pair, or not based on several identification parameters. The number of CMC pairs as a threshold is required so that the two images of surface topographies can be either identified as matching or determined to be non-matching. To reliably estimate error rates or evaluate likelihood ratio (LR), the key is to find an appropriate probability distribution for the frequency distribution of the observed CMC results. This paper discusses four statistical models for CMC measurements, which are binomial and three binomial-related probability distributions. In previous studies, for a sequence of binomial distributed or other binomial-related distributed random variables (r.v.), the number of Bernoulli trials N for each r.v. is assumed to be the same. However, in practice, N(the number of cell pairs in an image pair) varies from one r.v. (or one image pair) to another. In that case, the term, frequency function, of the CMC results is not appropriate. In this paper, the generalized frequency function is introduced to depict the behavior of the CMC values and its limiting distribution is provided. Based on that, nonlinear regression models are used to estimate the model parameters. The methodology is applied to a set of actual CMC values of fired cartridge cases.

This chapter begins with the elementary aspects of probability by starting with sets and Venn diagrams. The concepts of probability follow with appropriate descriptions of marginal, joint, conditional, and total probabilities, Bayes’ rule, Bernoulli trials, etc. Examples include those that examine the notion of continuous probability as a prelude to the presentation of random variables in the next chapter. Keeping with the theme of application-oriented content, the chapter contains topics in data analytics such as the estimation of a priori, conditional, and a posteriori probabilities associated with a given set of data collected from measurements. The presentation of the subject matter is organized to offer the reader the importance and relevance of the topics to present-day engineering problems. The association between transition matrix and confusion matrix is introduced to illustrate the connection of the probability concepts to data science. Examples and exercises include conceptual and data analytics-based ones.

A block-structured model for the reconstruction of directed and weighted financial networks, spanning multiple countries, is developed. In a first step, link-probability matrices are derived via a fitness model that is calibrated to reproduce a desired density and reciprocity for each block (i.e. country and cross-border sub-matrix). The resulting probability matrix allows for fast simulation through bivariate Bernoulli trials. In a second step, weights are allocated to a sampled adjacency matrix via an exponential random graph model (ERGM), which fulfills the row, column, and block weights. This model is analytically tractable, calibrated only on scarce publicly available data, and closely reconstructs known network characteristics of financial markets. In addition, an algorithm for the parameter estimation of the ERGM is presented. Furthermore, calibrating our model to the EU interbank market, we are able to assess the systemic risk within the European banking network by applying various contagion models.

This paper models age replacement and block replacement when there is the possibility of defaulting on the planned maintenance. A default occurs when a planned preventive replacement is not executed, and we discuss how defaults can arise in practice. Our aim is to study the robustness of block replacement and age replacement, bearing in mind that (a) these policies are frequently used in practice, (b) in the standard scenario (no defaulting) age replacement has a lower economic cost rate than block-replacement and (c) block replacement is simple to manage because component age does not have to be monitored. We model defaults as independent Bernoulli trials. We prove that a cost-minimizing critical age for replacement in the age policy with defaulting exists if the time to failure distribution has an increasing failure rate. A numerical study of the policies indicates that: age replacement is effective if maintenance control is good, that is, when there is only a small chance of defaulting; block replacement is relatively robust to defaulting (postponement), but less so to lack of knowledge about component reliability.

International audience; In the paper, the authors extend a function arising from the Bernoulli trials in probability and involving the gamma function to its largest ranges, find logarithmically complete monotonicity of these extended functions, and, in light of logarithmically complete monotonicity of these extended functions, derive some inequalities for multinomial coefficients and multivariate beta functions. These results recover, extend, and generalize some known conclusions.

The traditional theory of reliability is based on the Bernoulli trials, i.e., either success or failure. However, it seems unrealistic for enormous complex systems like computer networks. To overcome this issue, in the present study, an effort has been made regarding the development of a mathematical model for a computer network system and analysis of fuzzy availability of the same. The concepts of constant failure, constant repair, and coverage factor have been used for the development of the model. Impact of coverage factor, repair rates and failure rates of components has been analyzed on fuzzy availability of system. Markov birth-death process has been used for the development of Chapman-Kolmogorov differential-difference equations. The leading differential equations have been simplified by Runge–Kutta method of order four employing MATLAB (Ode 45 function).

In this paper, the model of a production inventory system with heterogeneous servers, the vacation of one of the servers (vacationing server), and retrial customers is considered. The customers reach the system each demanding exactly one unit of the item, according to a Poisson process. Service times and lead time are independent and follow exponential distributions. A single unit of the item is produced at a time according to (s, S) policy. It is assumed that when the inventory level reaches zero or when the orbit is empty or both, one of the servers(server 2) takes multiple, exponentially distributed vacation time. At the end of the vacation, that server immediately takes another vacation, if he/she finds empty stock or empty orbit or both. During the stock out period or server busy period or vacation period, the demands that arrive enter into the orbit of infinite size. If a customer finds both the servers busy or server 1 busy and server 2 in vacation or inventory level is zero, in accordance with Bernoulli trials may wait in the orbit or leave forever. When either the customer in the orbit cannot make an attempt due to the busy servers or inventory level zero, he/she under Bernoulli trials may come back to the orbit or leave the orbit. The rate of retrial customers from orbit is linear. The production process is switched on when the inventory level reaches to s, and similarly when the level of inventory reaches back to S the production process is switched off. The production process releases items that have inter arrival times that are exponentially distributed. Ergodicity condition is obtained and matrix analytic method is used to calculate the steady-state probabilities of the constructed 4D Markov chain. Minimum expected total cost per unit time and many significant system performance measures are obtained. Sensitivity analysis is relevant to check various performance measures that is highly applicable in the realistic situation and it also provides managerial insights.

Binary phase noise waveform is difficult to detect due to its thrum nail ambiguity function, has high transmission efficiency due to its constant magnitude, and is simple for implementation. Because of these advantages, binary phase noise waveform is widely used in noise radar. Many techniques have been developed to design binary phase noise waveform including Bernoulli trial, logistic mapping function, neural network optimization, genetic algorithm, kicked rotor chaos, polyphase perturbing, and particle swarm optimization techniques. In this paper, we first discuss the characteristics that good binary phase noise waveform needs to have. Then we introduce Kolmogorov-Smirnov (KS) two sample test in nonparametric statistics and derive balanced random walk. From balanced random walk theory, we propose balanced random based approach for noise waveform design and demonstrate that the noise waveforms generated by balanced random walk approach have zero direct components. We implement the proposed approach for noise waveform design and show that the proposed method is over 10% better on maximum sidelobes, 30% better on sidelobe energy than Bernoulli trial noise waveform design and selection.

We introduce a new model for sums of exchangeable Bernoulli random variables. The proposed distribution is an approximation to the exact distributional form, and relies on the theory of completely monotone functions and the Laplace transform of a gamma distribution function. Using Monte Carlo methods, we show that this new model compares favorably to the beta-binomial model with respect to estimating the success probability of the Bernoulli trials and the correlation between any two variables in the exchangeable set. We apply the new methodology to two classic data sets and the results are summarized.

Since the industry standard approach to judge on the effectiveness of promotion is based on the impact on expected sales it cannot grasp other impacts in the distribution of future sales. Since retailers operate with very high strategic service level targets (e.g. 98%) high quantiles of the sales distribution matter more than expected sales, which calls for quantile regression. There are more merits from this approach than forecasting high quantiles: Using real-world data from a fashion retail store i show that the impact of promotion can turn from insignificant to significantly harmful. Choosing quantile regression requires special diagnostics. The quality of forecasting high quantiles should be measured by the implied stock outs. Ideally, the stock outs would form a Bernoulli trials process with probability 100% minus service level target (e.g. 2%). This can be tested with backtests from the risk management literature as is shown in a real-world case.

We have now developed the tools to derive some of the standard one dimensional discrete probability distributions. In this chapter, we will examine the binomial distribution and its limit, the Poisson distribution, which are two of the most common distributions we run across in applications.

We propose a new function estimator, called Wilson Score Kernel Density Estimation, that allows to esti-mate a mean probability and the surrounding confidence interval for parameterized processes with binomiallydistributed outcomes. Our estimator combines the advantages of kernel smoothing, from Kernel Density Esti-mation, and robustness to low number of samples, from Wilson Score. This allows for more robust and dataefficient estimates compared to the individual use of these two estimators. While our estimator is generallyapplicable for processes with binomially distributed outcomes, we will present it in the context of iterativeoptimization. Here we first show the advantage of our estimator on a mathematically well defined problem,and then apply our estimator to an industrial automation process.

The Shewhart Sign (\(\text {SN}\)) control chart is a well-known distribution-free statistical process monitoring tool due to its robustness to the violation of the normality assumption for observations. To the best of our knowledge, there is not yet a thorough understanding of what happens to the statistical properties of the \(\text {SN}\) control chart in the presence of observations tied to the monitored population quantile, for example, the median: this is an event occurring in practice, in particular when the process runs in-control, because of the measurement device resolution, which inevitably introduces a rounding-off error. In this paper, we tackle the problem and show that when ties occur, the Shewhart \(\text {SN}\) control chart is no longer distribution-free, even in the presence of a small probability of having ties. To solve the problem, we discuss some procedures to handle the occurrence of ties. The study shows that the best strategy simply consists in implementing a Bernoulli trial approach: in practice, ties are reconsidered by \(50\%\) chance as being greater or smaller than the monitored population quantile. We quantitatively show that this approach allows the distribution-free properties of the Shewhart \(\text {SN}\) to be generally preserved.

Feller (1945) provided a coupling between the counts of cycles of various sizes in a uniform random permutation of $[n]$ and the spacings between successes in a sequence of $n$ independent Bernoulli trials with success probability $1/n$ at the $n$th trial. Arratia, Barbour and Tavare (1992) extended Feller’s coupling, to associate cycles of random permutations governed by the Ewens $(\theta )$ distribution with spacings derived from independent Bernoulli trials with success probability $\theta /(n-1+\theta )$ at the $n$th trial, and to conclude that in an infinite sequence of such trials, the numbers of spacings of length $\ell $ are independent Poisson variables with means $\theta /\ell $. Ignatov (1978) first discovered this remarkable result in the uniform case $\theta = 1$, by constructing Bernoulli $(1/n)$ trials as the indicators of record values in a sequence of i.i.d. uniform $[0,1]$ variables. In the present article, the Poisson property of inhomogeneous Bernoulli spacings is explained by a variation of Ignatov’s approach for a general $\theta >0$. Moreover, our approach naturally provides random permutations of infinite sets whose cycle counts are exactly given by independent Poisson random variables.

We study p-Faulty Search, a variant of the classic cow-path optimization problem, where a unit speed robot searches the half-line (or 1-ray) for a hidden item. The searcher is probabilistically faulty, and detection of the item with each visitation is an independent Bernoulli trial whose probability of success p is known. The objective is to minimize the worst case expected detection time, relative to the distance of the hidden item to the origin. A variation of the same problem was first proposed by Gal [29] in 1980. Alpern and Gal [3] proposed a so-called monotone solution for searching the line (2-rays); that is, a trajectory in which the newly searched space increases monotonically in each ray and in each iteration. Moreover, they conjectured that an optimal trajectory for the 2-rays problem must be monotone. We disprove this conjecture when the search domain is the half-line (1-ray). We provide a lower bound for all monotone algorithms, which we also match with an upper bound. Our main contribution is the design and analysis of a sequence of refined search strategies, outside the family of monotone algorithms, which we call t-sub-monotone algorithms. Such algorithms induce performance that is strictly decreasing with t, and for all \(p \in (0,1)\). The value of t quantifies, in a certain sense, how much our algorithms deviate from being monotone, demonstrating that monotone algorithms are sub-optimal when searching the half-line.

In this paper, a simple and general method based on the finite Markov chain imbedding technique is proposed to determine the exact conditional distributions of runs and patterns in a sequence of Bernoulli trials given the total number of successes. The idea is that given the total number of successes, the Bernoulli trials are viewed as random permutations. Then, we extend the result to multistate trials. The conditional distributions studied here lead to runs and patterns-type distribution-free tests whose applications are widespread. Two applications are considered. First, a distribution-free test for randomness is applied to rainfall data at Oxford from 1858 to 1952. The second application is to develop runs and patterns-type distribution-free control charts which can be used as Phase I and/or Phase II control charts. Numerical results for two commonly used runs-type statistics, the longest run and scan statistics, are also given.

The distribution of ( k 1 , k 2 ) -runs is well-known (Dafnis et al., 2010), under independent and identically distributed (i.i.d.) setup of Bernoulli trials but is intractable under non i.i.d. setup. Hence, it is of interest to find a suitable approximate distribution for ( k 1 , k 2 ) -runs, under non i.i.d. setup, with reasonable accuracy. In this paper, pseudo-binomial approximation to ( k 1 , k 2 ) -runs is considered using total variation distance. The approximation results derived are of optimal order and improve the existing results.

Daily production target setting based on machine capacities and available wafer-in-process (WIP) is an important practice in a semiconductor fab characterized by re-entrant process steps sharing individual machine group capacities. Operations of individual machine groups will track their respective target guidance through detailed machine allocation and lot dispatching (MALD), inducing variations of wafer flow. Existing approaches do not explicitly address such target-induced variations (TIV), and the resultant target setting may incur throughput loss and prolonged cycle times. A novel design of target-setting algorithm considering TIV called TaTIV is proposed to take TIV into account for setting daily targets systematically. TaTIV integrates three innovations: i) a Bernoulli trial model for approximating TIV of MALD at a process step under a given target, ii) a hybrid and recursive tandem queue approximation of multiple-step target-induced wafer flows and flow times given initial WIPs and machine group capacities of the day, and iii) a fixed-point iteration between target setting and target-induced wafer flow estimation. Analyses and simulations exploiting fab data show that TaTIV sets targets closer to what are actually achieved at individual steps than those set without considering TIV, reduces inter-step variations of machine allocation, and improves fab throughput and cycle times.

Author(s): Papalexiou, SM; Markonis, Y; Lombardo, F; AghaKouchak, A; Foufoula-Georgiou, E | Abstract: Hydroclimatic variables such as precipitation and temperature are often measured or simulated by climate models at coarser spatiotemporal scales than those needed for operational purposes. This has motivated more than half a century of research in developing disaggregation methods that break down coarse-scale time series into finer scales, with two primary objectives: (a) reproducing the statistical properties of the fine-scale process and (b) preserving the original coarse-scale data. Existing methods either preserve a limited number of statistical moments at the fine scale, which is often insufficient and can lead to an unrepresentative approximation of the actual marginal distribution, or are based on a limited number of a priori distributional assumptions, for example, lognormal. Additionally, they are not able to account for potential nonstationarity in the underlying fine-scale process. Here we introduce a novel disaggregation method, named Disaggregation Preserving Marginals and Correlations (DiPMaC), that is able to disaggregate a coarse-scale time series to any finer scale, while reproducing the probability distribution and the linear correlation structure of the fine-scale process. DiPMaC is also generalized for arbitrary nonstationary scenarios to reproduce time varying marginals. Additionally, we introduce a computationally efficient algorithm, based on Bernoulli trials, to optimize the disaggregation procedure and guarantee preservation of the coarse-scale values. We focus on temporal disaggregation and demonstrate the method by disaggregating monthly precipitation to hourly, and time series with trends (e.g., climate model projections), while we show its potential to disaggregate based on general nonstationary scenarios. The example applications demonstrate the performance and robustness of DiPMaC.

The paper deals with three generalized dependent setups arising from a sequence of Bernoulli trials. Various distributional properties, such as probability generating function, probability mass function and moments are discussed for these setups and their waiting time. Also, explicit forms of probability generating function and probability mass function are obtained. Finally, two applications to demonstrate the relevance of the results are given., Comment: 15 pages

A well known result in probability is the Poisson limit for rare independent Bernoulli trials. The asymptotic result also holds for a multinomial setting where events are dependent. An interesting data type is a survival process in which one observes many individuals over time periods. The risk set is all those available at the beginning of each time period, thus the same individual can appear in successive risk sets. Individuals exit rarely. In our motivating example these are corporations who exit a public trading system by default or merger, so there are several exit types, hence the multinomial setting. There are also covariates available at the beginning of each period. We study the numbers of exits over time. Under rare multinomial conditions we show that the exits types converge to independent Poisson laws with respect to the exit types and also with respect to time. An immediate application is the construction of one step ahead predictions which may then be tabulated or plotted, giving a convenient tool to study themodel behaviour with respect to time. Thus one can obtain one step ahead prediction intervals for the number of exits of each type, in our case bankruptcy or merger. This is a tool that is useful for large institutional investors such as pension plans.

The impetus of this research is to present a generalized Bernoulli Trial collision scheme in the context of the direct simulation Monte Carlo (DSMC) method. Previously, a subsequent of several collision schemes have been put forward, which were mathematically based on the Kac stochastic model. These include Bernoulli Trial (BT), Ballot Box (BB), Simplified Bernoulli Trial (SBT) and Intelligent Simplified Bernoulli Trial (ISBT) schemes. The number of considered pairs for a possible collision in the above-mentioned schemes varies between N ( l ) ( N ( l ) − 1 ) / 2 in BT, 1 in BB, and ( N ( l ) − 1 ) in SBT or ISBT, where N ( l ) is the instantaneous number of particles in the l th cell. Here, we derive a generalized form of the Bernoulli Trial collision scheme (GBT) where the number of selected pairs is any desired value smaller than ( N ( l ) − 1 ), i.e., N sel ( N ( l ) − 1 ) , keeping the same the collision frequency and accuracy of the solution as the original SBT and BT models. We derive two distinct formulas for the GBT scheme, where both formula recover BB and SBT limits if N sel is set as 1 and N ( l ) − 1 , respectively, and provide accurate solutions for a wide set of test cases. The present generalization further improves the computational efficiency of the BT-based collision models compared to the standard no time counter (NTC) and nearest neighbor (NN) collision models.

We investigate the predictive capability of mathematical models of the type-token relationship applied to the vocabulary growth profiles of selected English language documents. We compare the existing Good-Toulmin and Heaps formulae with an alternative approach based on Bernoulli trial word selection from a fixed finite vocabulary using the Zipf and Zipf-Mandelbrot probability distributions. We make two major observations: firstly, while the Zipf-Mandelbrot model makes better predictions of vocabulary growth than the Zipf model, the optimized parameters of the latter correlate better than those of the former with statistics gleaned independently from the data. Secondly, the mean of the Zipf-Mandelbrot, Good-Toulmin and Heaps models provides a more consistent and unbiased prediction of vocabulary than any individual model alone.

In many contexts with endogenous physical risks – e.g., households, neighbourhood traffic calming, production quality control – risk reduction is a local public good. Risk-reduction incentives then depend on the protected population’s size. Focusing on a household’s physical risks modelled as an i.i.d. Bernoulli trials sequence with endogenous “success” probability, I give sufficient conditions for safety to increase with the number protected via both monotone comparative statics methodology and a “first-order” approach. I utilise a recursive decomposition of a covariance involving a monotonic function of a binomial variable and first-degree stochastic dominance (FSD). Because “protection” problems are generally non-concave, I give a detailed treatment of the second-order condition, again via FSD.

A sequence of independent Bernoulli trials, each of which is a success with probability p , is conducted. For k ∈ Z + , let X k be the number of trials required to obtain k consecutive successes. Using techniques from elementary probability theory, we present a derivation which ultimately yields an elegant expression for the probability mass function of X k , and is simpler in comparison to what is found in the literature. Following this, we use our derived formula to obtain explicit closed-form expressions for the complementary cumulative distribution function and the n th factorial moment of X k .

Certifying performance of mechanical components with experimental tests is time consuming and expensive, which motivates the development of efficient approaches for predicting the outcomes from such testing. We propose two methods based on Gaussian processes (GP) to estimate the probability that new components will pass future certification tests, while assessing our prediction confidence. The first method processes a set of Bernoulli trials into a suitable machine learning dataset and subsequently infers the probability of performing satisfactorily for new components using heteroscedastic bounded GP regression. The second method uses GP classification with linear kernels. We demonstrate that linear kernels are well suited for datasets representing snapshots of mechanical system responses by accurately reproducing the underlying physical trends in the data. This yields consistent probabilities of passing and provides high labeling accuracy, even with small datasets. We demonstrate these techniques on synthetic datasets consistent with ship cabinet certification tests. We achieve up to 100% accuracy using all of the training data, and at least 92% with only 10% of the available data. With a corrupted training set, we obtain at least 93% accuracy. In the regression framework, we demonstrate that introducing heteroscedasticity helps achieve significantly better accuracy than frequentist machine learning methods.