Clarifying the Language of Chance Using Basic Conditional Probability Reasoning: The Monty Hall Problem

Clarity and preciseness in the use of language is crucial when communicating mathematical and probabilistic ideas. Lack of these can make even the simplest problem difficult to understand and solve. One such problem is the Monty Hall problem. In the past, a controversy was stirred among professional mathematicians when trying to reach a consensus on a solution to the problem. The problem still creates confusion among some of those who are asked to solve it for the first time. We purport to demonstrate the use of more precise language of basic conditional probability could have prevented the controversy.

Conditional Probability Approach for Fault Detection in Photovoltaic Energy Farms

Detection of electric faults in photovoltaic (PV) farms enhances a sustainable service continuity of farm energy generation. In this paper, a probabilisticfunction is introduced to detect the faults in the PV farms. The conditional probability functions are adopted to detect different fault conditions such as internalstring faults, string-to-string faults, and string-to-negative terminal faults. As thediodes are important to make the PV farms in-service safely during the faults,the distribution currents of these faults are evaluated with different concepts ofdiode consideration as well as without considering any diode installation. Thispart of the study enhances the diode utilization in the PV farms concerning theprotection point of view. The PV string currents are used as inputs to the conditional probability detection algorithms. However, the setting of the fault detectiontechnique is not portable for the other PV systems due to broad ranges of PV system ratings. To accordingly generalize the proposed fault detection algorithm, thePV string currents are first normalized to the total array current for universallyapplying the detection function at different PV string ratings. The limiting faultresistances are evaluated to show the sensitivity of the proposed fault detector.The results ensure the application of the proposed probabilistic detection functionfor PV farms.

A Hierarchy of Compatibility and Comeasurability Levels in Quantum Logics with Unique Conditional Probabilities

In the quantum mechanical Hilbert space formalism, the probabilistic interpretation is a later ad-hoc add-on, more or less enforced by the experimental evidence, but not motivated by the mathematical model itself. A model involving a clear probabilistic interpretation from the very beginning is provided by the quantum logics with unique conditional probabilities. It includes the projection lattices in von Neumann algebras and here probability conditionalization becomes identical with the state transition of the Lueders-von Neumann measurement process. This motivates the definition of a hierarchy of five compatibility and comeasurability levels in the abstract setting of the quantum logics with unique conditional probabilities. Their meanings are： the absence of quantum interference or influence, the existence of a joint distribution, simultaneous measurability, and the independence of the final state after two successive measurements from the sequential order of these two measurements. A further level means that two elements of the quantum logic （events） belong to the same Boolean subalgebra. In the general case, the five compatibility and comeasurability levels appear to differ, but they all coincide in the common Hilbert space formalism of quantum mechanics, in von Neumann algebras, and in some other cases.

On Conditional Probabilities of Factoring Quadratics

Factoring quadratics over Z is a staple of introductory algebra and textbooks tend to create the impression that doable factorizations are fairly common. To the contrary, if coefficients of a general quadratic are selected randomly without restriction, the probability that a factorization exists is zero. We achieve a specific quantification of the probability of factoring quadratics by taking a new approach that considers the absolute size of coefficients to be a parameter n. This restriction allows us to make relative likelihood estimates based on finite sample spaces. Our probability estimates are then conditioned on the size parameter n and the behavior of the conditional estimates may be studied as the parameter is varied. Specifically, we enumerate how many formal factored expressions could possibly correspond to a quadratic for a given size parameter. The conditional probability of factorization as a function of n is just the ratio of this enumeration to the total number of possible quadratics consistent with n. This approach is patterned after the well-known case where factorizations are carried out over a finite field. We review the finite field method as background for our method of dealing with Z [x]. The monic case is developed independently of the general case because it is simpler and the resulting probability estimating formula is more accurate. We conclude with a comparison of our theoretical probability estimates with exact data generated by a computer search for factorable quadratics corresponding to various parameter values.

A Fast Product of Conditional Reduction Method for System Failure Probability Sensitivity Evaluation

Systemreliability sensitivity analysis becomes difficult due to involving the issues of the correlation between failure modes whether using analytic method or numerical simulation methods.A fast conditional reduction method based on conditional probability theory is proposed to solve the sensitivity analysis based on the approximate analytic method.The relevant concepts are introduced to characterize the correlation between failure modes by the reliability index and correlation coefficient,and conditional normal fractile the for the multi-dimensional conditional failure analysis is proposed based on the two-dimensional normal distribution function.Thus the calculation of system failure probability can be represented as a summation of conditional probability terms,which is convenient to be computed by iterative solving sequentially.Further the system sensitivity solution is transformed into the derivation process of the failure probability correlation coefficient of each failure mode.Numerical examples results show that it is feasible to apply the idea of failure mode relevancy to failure probability sensitivity analysis,and it can avoid multi-dimension integral calculation and reduce complexity and difficulty.Compared with the product of conditional marginalmethod,a wider value range of correlation coefficient for reliability analysis is confirmed and an acceptable accuracy can be obtained with less computational cost.

Testing conditional independence with data missing at random

It is known that conditional independence is a quite basic assumption in many fields of statistics. How to test its validity is of great importance and has been extensively studied by the literature. Nevertheless, all of the existing methods focus on the case that data are fully observed, but none of them seems having taken into account of the scenario when missing data are present. Motivated by this, this paper develops two testing statistics to handle such a situation relying on the idea of inverse probability weighted and augmented inverse probability weighted techniques. The asymptotic distributions of the proposed statistics are also derived under the null hypothesis. The simulation studies indicate that both testing statistics perform well in terms of size and power.

The feature on the posterior conditional probability of finite state Markov channel

The feature of finite state Markov channel probability distribution is discussed on condition that original I/O are known. The probability is called posterior condition probability. It is also proved by Bayes formula that posterior condition probability forms stationary Markov sequence if channel input is independently and identically distributed. On the contrary, Markov property of posterior condition probability isn’t kept if the input isn’t independently and identically distributed and a numerical example is utilized to explain this case. The properties of posterior condition probability will aid the study of the numerical calculated recurrence formula of finite state Markov channel capacity.

Monte Carlo numerical simulation and its application in probability analysis of long span bridges

To get the probability of long span bridges under the influence of external random factors, the Monte Carlo method using Latin hypercube sampling is applied. Combined with the condition assessment system on Runyang Suspension Bridge, which is the longest suspension bridge in China, the structural probabilities in normal and damaged situations are calculated with the external random factors considered including environmental temperature, wind load, load of vehicles, etc. The main assessment items contain the maximal vertical displacement of girder, the maximal stress of cables, the maximal horizontal displacement of towers etc. Finally, the probabilities and their cumulative distribution functions are provided. The analysis results can be plotted on line in a clear and vivid way, so the efficiency of assessment is increased and the decision-making of maintenance is more objective and accurate.

The Survival Probability in Generalized Poisson Risk Model

In this paper we generalize the aggregated premium income process from a constant rate process to a poisson process for the classical compound Poinsson risk model,then for the generalized model and the classical compound poisson risk model ,we respectively get its survival probability in finite time period in case of exponential claim amounts.

Controller design for stochastic nonlinear systems with matched conditions

This paper is concerned with the global boundedness problem for a class of stochastic nonlinear systems with matched conditions. The uncertainties in the systems are due to parameter variations and external stochastic disturbance. Only the matched conditions and the possible bound of the uncertainties are demanded. Based on the stochastic Lyapunov stability theory, an explicit controller is constructed in the gradient direction, which renders responses of the closed-loop systems be globally bounded in probability. When the systems degrade to linear systems, the controller becomes linear. Illustrative examples are given to show the effectiveness of the proposed method.

Quantitative assessment of flight safety under atmospheric icing conditions

A quantitative assessment method is proposed to sense the specific effects of atmospheric icing conditions on flight safety. A six degree-of-freedom computational flight dynamics model is used to study the effects of ice accretion on aircraft dynamics, and a pilot model is also involved. In order to investigate icing severity under different icing conditions, support vector regression is applied in establishing relationship between aircraft icing parameter and weather conditions. Considering the characteristics of aircraft icing accidents, a risk probability assessment model optimized by the particle swarm method is developed to measure the safety level. In particular, angle of attack is chosen as a critical parameter in this method. Results presented in the paper for a series of simulation show that this method captures the basic effects of atmospheric icing conditions on flight safety, which may provide an important theoretical reference for icing accidents avoidance.

Method for acquiring part load distribution coefficient of air conditioning system

This paper presents a method to acquire runtime distribution ratio of building air conditioning system under part load condition (part load coefficient of system) through practical energy consumption data. By utilizing monthly energy consumption data of the entire year as the analysis object,this paper identifies data distribution,verifies distribution characteristics and analyzes distribution probability density for the issue of running time distribution ratio of air conditioning system in part load zones in the whole operation period,thus providing a basic calculation basis for an overall analysis of energy efficiency of air conditioning system. In view of the general survey of public building energy consumption carried by the government of Chongqing,this paper takes the governmental office building as an example,the part load ratio coefficient corresponding to practical running of air conditioning system of governmental office building in Chongqing is obtained by utilizing the above probability analysis and the solving method of probability density function. By utilizing the ratio coefficient obtained using this method,the part load coefficient with any running ratio of air conditioning system can be obtained according to the requirement of analysis,which can be used in any load ratio for analyzing running energy efficiency of air conditioning system.

Nonparametric inferences for kurtosis and conditional kurtosis

Under the assumption of strictly stationary process, this paper proposes a nonparametric model to test the kurtosis and conditional kurtosis for risk time series. We apply this method to the daily returns of S＆P500 index and the Shanghai Composite Index, and simulate GARCH data for verifying the efficiency of the presented model. Our results indicate that the risk series distribution is heavily tailed, but the historical information can make its future distribution light-tailed. However the far future distribution＇s tails are little affected by the historical data.

Computational Precision of the Power Function for Conditional Tests of Assumptions of the Rasch Model

Draxler and Zessin [1] derived the power function for a class of conditional tests of assumptions of a psychometric model known as the Rasch model and suggested an MCMC approach developed by Verhelst [2] for the numerical approximation of the power of the tests. In this contribution, the precision of the Verhelst approach is investigated and compared with an exact sampling procedure proposed by Miller and Harrison [3] for which the discrete probability distribution to be sampled from is exactly known. Results show no substantial differences between the two numerical procedures and quite accurate power computations. Regarding the question of computing time the Verhelst approach will have to be considered much more efficient.

Climate Change, Variability and Rainfall Probability for Crop Planning in Few Districts of Central India

Trend and climate change were studied in annual rainfall data for the period of 62 years (1949-2010) of Sagar and 65 years (1945-2010) of Damoh districts of Bundelkhand region of central India. The analysis of weekly rainfall probability was also carried out at both the locations for field operations and crop planning in rainfed agricultural system for improving the farmer’s livelihood. Mean annual rainfall was 1233.8 mm with 32.6% variability for Sagar and 1225.1 mm with 30.3% variability for Damoh region. Sagar and Damoh region exhibited significant decreasing trend in the total amount of annual rainfall in the last 15 years (1996 to 2010). Also, both locations showed significant long-term decreasing trend in annual rainfall. Climate of both the regions have changed as in the recent 15 years annual rainfall had decreased by 156.4 and 310.7 mm at Sagar and Damoh from their long period average (LPA), respectively. The Initial and conditional rainfall probability analysis at Damoh reinforced that Initial probabilities {P (W)} of getting 10 mm rainfall per week was 76% during 25th (18-24 June) SMW. Thus, the seed bed preparation could be initiated during this week. Initial as well as conditional probability of wet week followed by wet week {P(W/W} of getting 20 mm rainfall was more than 80% during 27th SMW (2-7 July) in Sagar district. Therefore, this week is most suitable for sowing operation in this district.

G-理论下的欧式向下敲出看涨期权定价

用G-几何布朗运动描述标的股票价格变化,假设波动率属于某一确定区间,通过G-Girsanov变换得到风险中性测度下欧式向下敲出看涨期权的定价区间。实验对比了布朗运动与G-布朗运动的路径,讨论了G-二次变差的参数对期权价格区间的影响,并对期权价格进行敏感性分析,验证了模型的合理性。

Measuring Real Capital Adequacy in Extreme Economic Conditions： An Examination of the Swiss Banking Sector

The global financial crisis （GFC） has placed the creditworthiness of banks under intense scrutiny. In particular, capital adequacy has been called into question. Current capital requirements make no allowance for capital erosion caused by movements in the market value of assets. This paper examines default probabilities of Swiss banks under extreme conditions using structural modeling techniques. Conditional Value at Risk （CVaR） and Conditional Probability of Default （CPD） techniques are used to measure capital erosion. Significant increase in Probability of Default （PD） is found during the GFC period. The market asset value based approach indicates a much higher PD than external ratings indicate. Capital adequacy recommendations are formulated which distinguish between real and nominal capital based on asset fluctuations.

Study on the Recurrence Probability of Strong Earthquakes of Faults

Based on the physical model of Brownian passage time,the probabilities of recurrence of strong earthquakes on the major active faults in China are calculated in different predictive time spans,based mainly on the analysis of the earthquake preparation process before a strong earthquake occurs. Furthermore,the seismic risks on active faults are studied. The results show that the earthquake probabilities on the Xianshuihe fault,the Altyn Tagh fault,the east Kunlun fault and Xiaojiang fault are significantly greater than other faults in the Chinese mainland,which indicates that the level of stress accumulation on these faults are higher than on other faults. Therefore,these faults may have a seismic risk for strong earthquake in future.

Metallogeny and Emplacement Conditions of Continental Terminal 3 (Ct3) Iron Formations of the Niamey Region (Western Niger)

This study aims to characterize the different lithofacies of the Ct3 formation in the Niamey region, and to determine the distribution of major and trace elements, in order to highlight the conditions for the establishment of iron mineralization. A lithological column, synthesizing sections of selected outcrops in the vicinity of Niamey, was produced. The chemical compositions of the selected samples were determined by X-ray fluorescence (XRF) spectrometry. Microscopic analysis of the thin sections determined the gœthitic nature of the oolitic iron ore. The oolites show a quartz, limonitic or gœthitic nucleus. Sometimes the nucleus is absent. From a morphoscopic point of view, two types of oolites have been distinguished: spherical-shaped and ellipsoidal-shaped oolites. The oolites are either contiguous or disseminated, as the case may be, in a limonitic to goethitic cement or in a fine sandstone matrix. The larger oolites (pisolites) are relatively friable. They reflect the influence of a relatively turbulent to submerged environment. The hardground of the iron mineralized horizons are covered by quartz grains. They are indicative of a submerged or emergent environment. X-ray fluorescence analysis shows high Fe2O3 contents (50% to 80%) and variable contents of major elements SiO2, Al2O3, TiO2, MnO, MgO, CaO, K2O and P2O5 associated with certain trace elements such as Th, U, V, Y, Zn, Zr and As. The results of the study are an important tool for decision-makers to adopt effective prevention/remediation measures for groundwater contamination in the Continental terminal aquifer systems.

Determination of Dynamic Load Condition of Aircraft Carrier based on Multivariable Distribution of Flight Parameters

In order to solve the design problem of dynamic load of a carrier-based aircraft,according to the relevant criteria of national military standards,eight key flight parameters of the dynamic load conditions for the aircraft carrier were selected.Based on the multivariable distribution data of landing flight parameters obtained from flight test,the distribution form and range limitation of each flight parameter were determined by using the probability distribution of each flight parameter and the spatial relationship among variables.Furthermore,100000 sets of data were constructed to simulate the landing condition of aircraft in the form of random number.After the envelope is screened by multivariable joint probability distribution,the boundary conditions were compared and merged,and finally the dynamic load conditions were obtained.In this paper,a set of dynamic load condition design method based on the aircraft requirements is constructed,which systematically covers all kinds of situations that occur in the process of aircraft landing,and improves the conventional design process of aircraft dynamic load.