Theoretical Probability Research Articles

Developing the role of modelling in the teaching and learning of probability

ABSTRACT We examine the challenges of teaching probability through the use of modelling. We argue how an integrated modelling approach might facilitate a coordinated understanding of distribution by marrying theoretical and data-oriented perspectives and present probability as more connected to the social lives of modern-day students. Research is, however, also showing that learning modelling is non-trivial. For example, students can easily be confused between real-world data and model-generated data. Furthermore, students tend to create models that reflect the structure of the problem context rather than its mathematical structure. The argument we present is that teaching probability through an integrated modelling approach could offer important educational benefits, but this would be an immense challenge to the skills and knowledge of teachers and would demand increased levels of resources for teacher development. The implication is that research needs to identify suitable forms of teacher development and designs for tasks and curricula.

Toplum Kendini Savunur: Biyopolitikanın Covid-19 Çıkmazı, Hayatın Tekilselliği ve Toplumsal Kara Delikler

This essay focuses on the existential conflict between society and politics that has once again come to the fore in the sense of biopolitical theory with the Covid-19 outbreak. It attempts to demonstrate where and how the theory of biopolitics is insufficient at understanding the individual, who has reset all relationships under the circumstances of the pandemic, while providing a certain viewpoint based on true and logical facts. The essay considers this insufficiency as a common deficiency of the various types of biopolitical theory and correlates this to a historical deficiency that has lacked a concept of absolute singularity (i.e., singularity of existence) that would precede the compromises and necessities associated with life in identifying socio-political origins. Thus, the article argues the theory of biopolitics to have persisted in this sense within the theories of classical order, particularly the Hobbesian theory of social contract, based on replicated historical deficiency. Therefore, the key issue of the essay is whether a non-relational moment of existence exists for any nature or framework that refutes all the responsibilities, concessions, or regularities attached to it such that it cannot be appropriated. The essay affirms this issue through a theoretical probability and attempts to view the circumstances present in the Covid-19 phase not as the moment itself but as its signals, messages, and indications. This moment is conceptualized as a social black hole. Thus, the essay examines the destructive and constitutive role of the moment in which life sinks into social black holes.

Multiple Complex Symbol Golden Space-Time Labeling Diversity

The Golden code is a full-rate space-time block code (STBC), which achieves a diversity order of $2N_{r}$ , where $N_{r}$ is the number of receive antennas. The multiple complex symbol Golden code (MCS-Golden code) is a generalized Golden code. The MCS-Golden code not only achieves full-rate but also achieves a diversity order of $2^{n}{N_{r}}$ , $n$ , $n\ge {1}$ , is an integer, and $2^{n}$ is the number of multiple complex symbols in the MCS-Golden code system. In this paper, we apply the labelling diversity technique to the MCS-Golden code to form a new type of STBC, named as multiple complex symbol Golden space-time labelling diversity (MCS-GSTLD). The proposed MCS-GSTLD achieves a diversity order of $2\times {2^{n}}{N_{r}}$ , and further improves error performance compared to the MCS-Golden code. We also formulate the theoretical average bit error probability (ABEP) for the proposed MCS-GSTLD system. Furthermore, we apply sphere decoding with sorted detection subset (SD-SDS) to detect the transmitted symbols in the MCS-GSTLD system. Simulation results show that the SD-SDS scheme achieves the theoretical ABEP for $N_{r}>2$ . Both theoretical and simulation results demonstrate that with the same spectral efficiency and four receive antennas, the 2CS-GSTLD achieves at least 1.3 dB signal-to-noise ratio (SNR) gain compared to the 2CS-Golden code at a BER of $3\times {10^{-6}}$ , while the 4CS-GSTLD achieves at least 0.7 dB SNR gain compared to the 4CS-Golden code.

Reconfigurable Intelligent Surface-Aided Single-Input Single-Output K-Complex Symbol Golden Codeword-Based Modulation

Small reconfigurable intelligent surfaces (RISs) have greater potential for flexibility and ubiquity compared to large RISs. In the literature, the theoretical average bit error probability (ABEP) of a phase-adjustable element access-point-based RIS (AP-RIS) is derived by exploiting the central-limit theorem for large surface size. Hence, the analysis does not hold for small surface sizes. In this paper, we first formulate a simple closed-form expression for the ABEP of a phase-adjustable element AP-RIS with $M$ -ary quadrature amplitude modulation ( $M$ QAM). Compared to the results available in literature, this expression is valid even for small RISs. Second, we investigate AP-RIS-aided single-input single-output $K$ -complex symbol Golden codeword-based modulation, which encodes/transmits $K~M$ QAM symbols in each of $K$ consecutive independently faded time slots. The encoding ensures that each symbol experiences $K$ different fadings, hence resulting in a $K$ -fold increase in diversity order, while the AP-RIS maximizes the received signal-to-noise ratio (SNR) in each slot and yields significant SNR gains with increasing size. An increase in $K$ yields significant SNR gains for small RISs. A theoretical bound on the ABEP is formulated and validated by simulation. Finally, a detector based on sorted symbol set sphere decoding is devised.

Selecting the probability distribution of annual maximum temperature in Malaysia

The issues on global warming have become very popular and been discussed both locally and internationally. This phenomenon due to the temperature rises will increase the variability of climate and more natural disasters were expected to occur. Increasing of global temperature will affect the agricultural sector, increase some of the infectious diseases that may lead to high mortality rates in humans, high demand for electricity, water and food which eventually affecting the economy of Malaysia. Hence, this work aims to study the best fitted probability distribution that describes the annual maximum temperature recorded at seventeen meteorological stations in Malaysia. The Normal, Lognormal, Gamma, Weibull and Generalized Skew Logistic distributions are considered using the maximum likelihood estimation method to estimate the parameters. The goodness of fit test and model selection criteria such as Kolmogorov-Smirnov and AndersonDarling tests, Corrected Akaike Information Criterion and Bayesian Information Criterion are used to measure the accuracy of the predicted data using theoretical probability distributions. The results show that most of the stations favour the Generalized Skew Logistic distribution as the best fitted probability distribution. Also, some stations favour the Normal, Lognormal as well as Weibull distribution as the best fitted distribution to describe the annual maximum temperature.

A New Approach to the Maximum Quarterly Water Consumption Modeling on the Example of Individual Water Consumers in a Small Water Supply System

Quarterly water consumption data collected in a small water supply system were used for elaboration of a new water consumption modeling approach. In this paper, multi-distribution statistical analysis was performed. As the Anderson-Darling test proved, at least a half out of the ten tested theoretical probability distributions can be used for description of the water consumption. The application of the PWRMSE criterion made it possible to determine, which of the tested theoretical distributions is the best-fitted to the empirical data set. In the case of total daily water consumption for the group of the households, it was Johnson distribution, whereas for the average daily water consumption per capita, it was GEV distribution. Based on the best-fitted probability distribution, a 25-year water consumption simulation with the Monte Carlo method was conducted. Because methodology of this study is based on the probability distributions, even if the type of theoretical distribution of the water consumption will change, it will be still possible to use this simulation method by assuming the other distribution.

ВЗАЄМОДІЯ ГРАВІТАЦІЙНИХ ХВИЛЬ З ПРИПЛИВНОЮ ЕЛЕКТРОСТАНЦІЄЮ

Використання відновлюваних джерел енергії є нагальним та актуальним у життєдіяльності людства в сучасних умовах паливо-енергетичної кризи та забруднення навколишнього середовища. Одним із типів відновлюваних джерел енергії є припливні електростанції, в яких використовується потенційна енергія припливів і відливів. Визначення особливостей взаємодії хвильової течії з конструкціями припливної електростанції та створення умов безаварійної та ефективної роботи таких станцій є, безумовно, актуальною проблемою в сучасній енергетиці. Метою досліджень є визначення впливу гравітаційних хвиль на конструкцію припливної електростанції та розроблення рекомендацій щодо безпечних умов експлуатації припливної електростанції. В роботі проводяться чисельні та експериментальні дослідження взаємодії хвильової течії зі спорудами припливної електростанції та дамбою, яка огороджує закриту акваторію. Для цього використано сучасний апарат теоретичної гідромеханіки, теорії ймовірностей та математичної статистики, створено експериментальний стенд і проведені лабораторні дослідження в хвильовому басейні, де генератор хвиль генерував гравітаційні хвилі заданих параметрів. Проведені візуальні дослідження і визначені інтегральні та спектральні характеристики полів швидкості та тиску. Отримані гідродинамічні характеристики хвильового руху та течій через турбінні тракти електростанції і їх спектральні складові. Визначені поля тиску, навантаження і перекидуючих моментів сил на будівлю станції та встановлені особливості розмиву ґрунту у місці спряження станції та дна водойми. Проведені оцінки характеристик розмиву піщаного ґрунту поблизу припливної електростанції. Розроблені рекомендації щодо безаварійної роботи припливної електростанції та визначені оптимальні товщини піщаної подушки та кам’яної берми, на яких будується станція. Бібл. 20, рис. 5.

Estimate the probability density function of maximum temperature for the Middle East

Pollution is one reasons for increase temperature which leads to increase the heat waves which have large socioeconomic and healthy impacts on Middle East. By using monthly daily mean of maximum temperature (C°) at height (2m) covered middle east as a grid of (1581) points for selected months (March, April, May) represent spring and (Jun, July, August) represent Summer for the period 1979 to2018, from the ECMWF, model ERA-interim. Many PDFs have been proposed in recent past, but in present study Logistic, Rayleigh and Gamma distribution are used to describe the characteristics of maximum temperature. This paper attempts to determine the best fitted probability distribution of maximum temperature. To check the accuracy of the predicted data using theoretical probability distributions the goodness of fit criteria Z-test used in this paper. According to the goodness-of-fit criteria and from the graphical comparisons it can be said that Logistic distribution provides the best fit for the observed monthly daily mean of maximum temperature data.

COPULA REPRESENTATIONS FOR THE SUM OF DEPENDENT RISKS: MODELS AND COMPARISONS

The study of the distributions of sums of dependent risks is a key topic in actuarial sciences, risk management, reliability and in many branches of applied and theoretical probability. However, there are few results where the distribution of the sum of dependent random variables is available in a closed form. In this paper, we obtain several analytical expressions for the distribution of the aggregated risks under dependence in terms of copulas. We provide several representations based on the underlying copula and the marginal distribution functions under general hypotheses and in any dimension. Then, we study stochastic comparisons between sums of dependent risks. Finally, we illustrate our theoretical results by studying some specific models obtained from Clayton, Ali-Mikhail-Haq and Farlie-Gumbel-Morgenstern copulas. Extensions to more general copulas are also included. Bounds and the limiting behavior of the hazard rate function for the aggregated distribution of some copulas are studied as well.

Probability distributions of daily rainfall extremes in Lazio and Sicily, Italy, and design rainfall inferences

Study regionWe investigate samples from two Italian regions, i.e. Lazio and Sicily, located in central and south Italy, respectively, and characterized by two diverse climates. Study focusIn engineering practice, the study of maxima daily rainfall values is commonly dealt with light-tailed probability distribution functions, such as the Gumbel. The choice of a distribution rather than another may cause estimation errors of rainfall values associated to specific return periods. Recently, several studies demonstrate that heavy-tailed distributions are preferable for extreme events modelling. Here, we opt for six theoretical probability distribution functions and evaluate their performance in fitting extreme precipitation samples. We select the samples with two common methods, i.e. the Peak-Over-Threshold and the Annual Maxima. We assess the best fitting distribution to the empirical samples of extreme values through the Ratio Mean Square Error Method and the Kolmogorov-Smirnov test. New hydrological insights for the regionThe assessment of the best fitting distribution to daily rainfall of the two different areas investigated here leads to interesting remarks. Despite the diversity of their climate, results suggest that heavy-tailed distributions describe more accurately empirical data rather than light-tailed ones. Therefore, extreme events may have been largely underestimated in the past in both areas. The proposed investigation can prompt the choice of the best fitting probability distribution to evaluate the design hydrological quantities supporting common engineering practice.

Low Complexity Adaptation for Reconfigurable Intelligent Surface-Based MIMO Systems

Reconfigurable intelligent surface (RIS)-based transmission technology offers a promising solution to enhance wireless communication performance cost-effectively through properly adjusting the parameters of a large number of passive reflecting elements. This letter proposes a cosine similarity theorem-based low-complexity algorithm for adapting the phase shifts of an RIS that assists a multiple-input multiple-output (MIMO) transmission system. A semi-analytical probabilistic approach is developed to derive the theoretical average bit error probability (ABEP) of the system. Furthermore, the validity of the theoretical analysis is supported through extensive computer simulations.

Development of whole-genome multiplex assays and construction of an integrated genetic map using SSR markers in Senegalese sole

The Senegalese sole (Solea senegalensis) is an economically important flatfish species. In this study, a genome draft was analyzed to identify microsatellite (SSR) markers for whole-genome genotyping. A subset of 224 contigs containing SSRs were preselected and validated by using a de novo female hybrid assembly. Overall, the SSR density in the genome was 886.7 markers per megabase of genomic sequences and the dinucleotide motif was the most abundant (52.4%). In silico comparison identified a set of 108 SSRs (with di-, tetra- or pentanucleotide motifs) widely distributed in the genome and suitable for primer design. A total of 106 markers were structured in thirteen multiplex PCR assays (with up to 10-plex) and the amplification conditions were optimized with a high-quality score. Main genetic diversity statistics and genotyping reliability were assessed. A subset of 40 high polymorphic markers were selected to optimize four supermultiplex PCRs (with up to 11-plex) for pedigree analysis. Theoretical exclusion probabilities and real parentage allocation tests using parent–offspring information confirmed their robustness and effectiveness for parental assignment. These new SSR markers were combined with previously published SSRs (in total 229 makers) to construct a new and improved integrated genetic map containing 21 linkage groups that matched with the expected number of chromosomes. Synteny analysis with respect to C. semilaevis provided new clues on chromosome evolution in flatfish and the formation of metacentric and submetacentric chromosomes in Senegalese sole.

A Novel Index Modulation Dimension Based on Filter Domain: Filter Shapes Index Modulation

A novel domain for Index Modulation (IM) named “Filter Domain” is proposed. This new domain generalizes many existing modulations and IM domains. In addition, a novel scheme “Filter Shape Index Modulation” (FSIM) is proposed. This FSIM scheme allows a higher Spectral Efficiency (SE) gain than the time and frequency IM dimensions in Single-Input Single-Output (SISO) systems. In the FSIM system, the bit-stream is mapped using an Amplitude Phase Modulation (APM) as QAM or PSK, and an index of a filter-shape changing at the symbol rate. This filter shape, being changed at each symbol, enables a SE gain in SISO system without sacrificing any time or frequency resources. Compared to an equivalent 8QAM and 16QAM schemes and at the same SE, the FSIM with QPSK using 2 and 4 non-optimal filter shapes achieves a gain of 3.8 dB and 1.7 dB respectively at BER= 10−4, and this superiority is maintained in frequency selective fading channel compared to equivalent SISO-IM schemes. A low complexity detection scheme, approaching the maximum likelihood detector performance, is proposed along with a full performance characterization in terms of theoretical probability of filter index error and BER lower bound. Finally, FSIM can achieve better spectral and energy efficiencies when a filter bank and an ISI cancellation technique are optimally designed.

The effect of liquid co-flow on gas fractions, bubble velocities and chord lengths in bubbly flows. Part I: Uniform gas sparging and liquid co-flow

Unique experiments were performed in a homogeneously sparged rectangular 400×200×2630 mm (W×D×H) bubble column with and without liquid co-flow. Bubbles in the range 4–7 mm were produced by needle spargers, which resulted in a very uniform bubble size. Dual-tip optical fibre probes were used to measure horizontal profiles of gas fractions, bubble velocities and bubble chord lengths for superficial gas velocities Usg in the range 0.63–6.25 cm/s and superficial liquid velocities Usl up to 20 cm/s. Images of the bubble column were captured and a Bubble Image Velocimetry technique was adopted to calculate bubble (parcel) velocities. For low gas fractions, when a homogeneous flow regime occurred, both methods agreed very well and the optical fibre probes were found to be rather accurate for our bubbles. A liquid co-flow was found to have a calming effect and to stabilize a homogeneous bubbly flow regime, with less spatial variation in gas fractions and bubble velocities. Bubble chord lengths were almost normally distributed and do not exhibit the theoretical triangular probability density functions. The mean cord lengths were in the range 1.9–3.5 mm and found to increase with Usg and to decrease slightly with increasing Usl, while a liquid co-flow significantly reduced the standard deviation of the chord length distribution.

Instar determination for sugarcane internode borer Chilo sacchariphagus indicus (Kapur) (Lepidoptera: Crambidae)

• Sugarcane internode borer (INB) passes through five larval instars to become a pupa. • Frequency distribution of head capsule widths is reliable in instar fixation in INB. • Dyar’s ratios and regression analyses confirmed that no instar has been overlooked. • The probability of misclassifying an INB instar into another is remote (0.16–1.97%). Internode borer (INB), Chilo sacchariphagus indicus has been a serious threat to sugarcane cultivation for more than six decades. We have determined the number of instars for INB through frequency distribution analysis of cast head capsule widths. This is the first report, which proves that the INB of sugarcane passes through five larval instars to attain its pupal stage. The number of instars determined by kernel density estimation was in corroboration with the number of instars observed in the laboratory. The mean Dyar’s ratios for both the instar-wise observed data on head capsule widths and the theoretical data derived from the frequency distribution analysis were one and the same (1.46). Linear regression (R 2 > 0.998) between the instar numbers and their corresponding mean head capsule widths reaffirmed that no instar has been overlooked. Further, the theoretical misclassification probabilities of 0.16–1.97% indicates that the chance of misidentifying an INB instar into its preceding or succeeding one is very remote. Since the determination of the exact number of instars in a pest species is a pre-requisite for developing appropriate management strategies, the outcome of this study holds great promise in managing the most notorious Crambid borer of sugarcane.

Concurrent sequences of Bernoulli trials

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.”

A universal Standardized Precipitation Index candidate distribution function for observations and simulations

Abstract. The Standardized Precipitation Index (SPI) is a widely accepted drought index. Its calculation algorithm normalizes the index via a distribution function. Which distribution function to use is still disputed within the literature. This study illuminates that long-standing dispute and proposes a solution that ensures the normality of the index for all common accumulation periods in observations and simulations. We compare the normality of SPI time series derived with the gamma, Weibull, generalized gamma, and the exponentiated Weibull distribution. Our normality comparison is based on a complementary evaluation. Actual compared to theoretical occurrence probabilities of SPI categories evaluate the absolute performance of candidate distribution functions. Complementary, the Akaike information criterion evaluates candidate distribution functions relative to each other while analytically punishing complexity. SPI time series, spanning 1983–2013, are calculated from the Global Precipitation Climatology Project's monthly precipitation dataset, and seasonal precipitation hindcasts are from the Max Planck Institute Earth System Model. We evaluate these SPI time series over the global land area and for each continent individually during winter and summer. While focusing on regional performance disparities between observations and simulations that manifest in an accumulation period of 3 months, we additionally test the drawn conclusions for other common accumulation periods (1, 6, 9, and 12 months). Our results suggest that calculating SPI with the commonly used gamma distribution leads to deficiencies in the evaluation of ensemble simulations. Replacing it with the exponentiated Weibull distribution reduces the area of those regions where the index does not have any skill for precipitation obtained from ensemble simulations by more than one magnitude. The exponentiated Weibull distribution maximizes also the normality of SPI obtained from observational data and a single ensemble simulation. We demonstrate that calculating SPI with the exponentiated Weibull distribution delivers better results for each continent and every investigated accumulation period, irrespective of the heritage of the precipitation data. Therefore, we advocate the employment of the exponentiated Weibull distribution as the basis for SPI.

An Analysis Method for Solving Ambiguity in Direction Finding with Phase Interferometers

The phase interferometer is an effective direction finding (DF) method. It is widely utilized in various electronic reconnaissance systems due to its advantages of fast operation and high precision. In a wideband system, the combination of long and short baselines, or even multi-level baselines, is applied in an interferometer to solve the contradiction between accuracy and phase ambiguity for DF. In this paper, a novel analysis method is proposed to obtain the probability of successfully solving ambiguity based on mathematical statistics. According to the length ratio between short and long baselines, the joint density function of phase errors can be derived. Then, the probability can be achieved under different signal-to-noise ratios (SNRs) by integrating the joint density function in a specific interval. Furthermore, the formula of phase measurement error is adjusted by the least square method to improve computational accuracy in low SNR during the process. Under different baseline configurations, the strategy can provide the theoretical probability of successfully solving ambiguity, thus guiding the baseline design for obtaining a maximum probability without impacting the specified DF accuracy. Simulation results show that the mathematical model is efficient in some complex cases.

The branching ratio of intercombination [formula omitted]transitions in the RbCs molecule: Measurements and calculations

We observed the A1Σ+∼b3Π→a3Σ+/X1Σ+laser-induced fluorescence (LIF) of the RbCs molecule excited from the ground X1Σ+state by the Ti:Sapphire laser. The LIF spectra from the common perturbed levels of the singlet-triplet A ∼ b complex was recorded by the Fourier-transform (FT) spectrometer with the instrumental resolution of 0.03 cm−1. The relative intensity distribution in the rotationally resolved A∼b→a3Σ+(va)/X1Σ+(vX)progressions was measured, and their branching ratio was found to be about of 1÷5×10−4in the bound region of the a3Σ+and X1Σ+states. The experiment was complemented with the scalar- and full-relativistic calculations of the A/b−X/atransition dipole moments (TDMs) as functions of internuclear distance. The relative systematic error in the resulting ab initio TDM functions evaluated for the strong A−Xtransition was estimated as few percent in the energy region, where the experimental LIF intensities are relevant. The relative spectral sensitivity of the FT registration system, operated with the InGaAs diode detector and CaF beam-splitter, was calibrated in the range [6 500, 12 000] cm−1by a comparison of experimental intensities in the long A ∼ b → X(vX) LIF progressions of the K2 and KCs molecules with their theoretical counterparts evaluated using the ab initioA−XTDMs. Both experimental and theoretical transition probabilities can be employed to improve the stimulated Raman adiabatic passage process, a → A ∼ b → X, which is exploited for a laser assembling of ultracold RbCs molecules.

Generalized Space Time Block Coded Spatial Modulation for Open-Loop Massive MIMO Downlink Communication Systems

In this paper, we propose a generalized space-time block coded spatial modulation (GSTBC-SM) scheme for open-loop massive multiple-input and multiple-output (MIMO) downlink communication systems. Specifically, we firstly partition the information bits into multiple groups with each group modulated by the spatial modulation (SM), where the SM symbols are invoked for orthogonal STBC (OSTBC) and quasi-orthogonal STBC (Q-OSTBC) structures. Then, message passing (MP) and block minimum mean square equalization (B-MMSE) detectors are designed for our GSTBC-SM systems, to achieve near-optimal performance with significantly reduced complexity in massive MIMO configurations. Finally, we derive the theoretical average bit error probability (ABEP) of the proposed scheme. The main contribution is that the propose scheme achieves high transmission rate and diversity gain even with small number of radio frequency (RF) chains at the transmitter. Simulation results verify the theoretical derivations and show that the proposed GSTBC-SM scheme provides near 20 dB gain over the conventional GSTBC scheme under massive MIMO configurations.