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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,...