Algorithms for operations on probability distributions in a computer algebra system

Algorithms for computing the distributions of sums of discrete random variables

Previous article Next article On the Extension of the Class of Stable DistributionsV. M. KruglovV. M. Kruglovhttps://doi.org/10.1137/1117081PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] B. V. Gnedenko and , A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, Mass., 1962 Google Scholar[2] Yu. V. Linnik, Decomposition of probability distributions, Edited by S. J. Taylor, Dover Publications Inc., New York, 1964xii+242 MR0189085 (32:6512) 0121.35502 Google Scholar[3] B. V. Gnedenko, Some theorems on distribution functions, Učenie Zapiski Moskovskogo Univ., 30 (1939), 61–82, (In Russian.) Google Scholar[4] V. M. Kruglov, A note on infinitely divisible distributions, Theory Prob. Applications, 15 (1970), 319–324 10.1137/1115038 0329.60011 LinkGoogle Scholar[5] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– MR0322926 (48:1287) 0219.60027 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Functional L.I.L. for the Upper Extremes of i.i.d. Random Variarbles from a Sub-class of Heavy Tailed Distributions31 October 2021 | Journal of the Indian Society for Probability and Statistics, Vol. 22, No. 2 Cross Ref Functional law of the iterated logarithm for partial sum processes of non-negative valued iid random variables belonging to the domain of partial attraction of a semi-stable lawStatistics & Probability Letters, Vol. 137 Cross Ref A Log Log Law for Subsequences of Delayed Random Sums27 July 2017 | Journal of the Indian Society for Probability and Statistics, Vol. 18, No. 2 Cross Ref Merging to Semistable ProcessesI. 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This chapter introduces probability theory as a system of models, based on measure theory, of some real-world phenomena. The models are measure spaces of total measure 1 and usually have certain distinguished measurable functions defined on them. Section 1 begins by establishing the measure-theoretic framework and a short dictionary for passing back and forth between terminology in measure theory and terminology in probability theory. The latter terminology includes events, random variables, mean, probability distribution of a random variable, and joint probability distribution of several random variables. An important feature of probability is that it is possible to work with random variables without any explicit knowledge of the underlying measure space, the joint probability distributions of random variables being the objects of importance. Section 2 introduces conditional probability and uses that to motivate the mathematical definition of independence of events. In turn, independence of events leads naturally to a definition of independent random variables. Independent random variables are of great importance in the subject and play a much larger role than their counterparts in abstract measure theory. Examples at the end of the section indicate the extent to which functions of independent random variables can remain independent. The techniques in the examples are of use in the subject of statistical inference, which is introduced in Section 10. Section 3 states and proves the Kolmogorov Extension Theorem, a foundational result allowing one to create stochastic processes involving infinite sets of times out of data corresponding to finite subsets of those times. A special case of the theorem provides the existence of infinite sets of independent random variables with specified probability distributions. Section 4 establishes the celebrated Strong Law of Large Numbers, which says that the Cesaro sums of a sequence of identically distributed independent random variables with finite mean converge almost everywhere to a constant random variable, the constant being the mean. This is a theorem that is vaguely known to the general public and is widely misunderstood. The proof is based on Kolmogorov’s inequality. Sections 5–8 provide background for the Central Limit Theorem, whose statement and proof are in Section 9. Section 5 discusses three successively weaker kinds of convergence for random variables—almost sure convergence, convergence in probability, and convergence in distribution. Convergence in distribution will be the appropriate kind for the Central Limit Theorem. Section 6 contains the Portmanteau Lemma, which gives some equivalent formulations of convergence in distribution, Section 7 introduces characteristic functions as Fourier transforms of probability distributions, and Section 8 proves the Levy Continuity Theorem, which formulates convergence in distribution in terms of characteristic functions. Section 9 contains the statement and proof of the Central Limit Theorem, followed by some simple examples. This theorem is the most celebrated result in probability theory and has many applications in mathematics and other fields. Section 10 is a brief introduction to the subject of statistical inference, showing how the Central Limit theorem plays a role in practice through the $t$ test of W. S. Gosset.

New concepts on fractional probability theory are introduced and some inequalities for the fractional weighted expectation and the fractional weighted variance of continuous random variables are obtained. Other fractional results related to the two orders-fractional % weighted moment are also established. Some recent results on integral inequality theory can be deduced as some special cases. At the end, some applications on the uniform random variable are given.

The concepts of variability and uncertainty came from experience and coexist with different connotations. First, the article reviews the statistical methods for variability assessments of probability distributions. Next, it sums up the entropy concept of uncertainty of systems of events in probability theory. The two concepts are brought closer together on the basis of common experience of predictability. The article also considers the concept of average number of equally probable events based on entropy. Then, it introduces the concept of equivalent number of outcomes based on variability of probability distributions. Finally, the link between variability and uncertainty is illustrated with examples.

On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables

This article details what resulted when I re-designed my undergraduate statistics course to allow failure as a learning strategy and focused on achievement rather than performance. A variety of within and between class t-tests are utilized to determine the impact of unlimited test and quiz opportunities on student learning on both quizzes and subsequent assignments in two undergraduate statistics courses (one a 15-week hybrid and one a 6-week online course). The results show that the previous DFW rate was reduced, and no significant difference in outcomes was found between the two different course modalities. Furthermore, students achieved significantly higher on their last quiz and test attempts on every instrument in both semesters, with large effect sizes. Corresponding assignments showed students achieved significantly higher than the mean first attempt, but significantly lower than final mean quiz attempt scores, showing that some knowledge was not carried over to application. The article concludes by evaluating the results of unlimited testing with minimum standards and the limitations of the study and the pedagogical model.

This thesis deals with recurrent neural networks, a particular class of artificial neural networks which can learn a generative model of input sequences. The input is mapped, through a feedback loop and a non-linear activation function, into a hidden state, which is then projected into the output space, obtaining either a probability distribution or the new input for the next time-step. This work consists mainly of two parts: a theoretical study for helping the understanding of recurrent neural networks framework, which is not yet deeply investigated, and their application to non-linear prediction problems, since recurrent neural networks are really powerful models suitable for solving several practical tasks in different fields. For what concerns the theoretical part, we analyse the weaknesses of state-of-the-art models and tackle them in order to improve the performance of a recurrent neural network. Firstly, we contribute in the understanding of the dynamical properties of a recurrent neural network, highlighting the close relation between the definition of stable limit cycles and the echo state property of an echo state network. We provide sufficient conditions for the convergence of the hidden state to a trajectory, which is uniquely determined by the input signal, independently of the initial states. This may help extend the memory of the network and increase the design options for the network. Moreover, we develop a novel approach to address the main problem in training recurrent neural networks, the so-called vanishing gradient problem. Our new method allows us to train a very simple recurrent neural network, making the gradient not to vanish even after many time-steps. Exploiting the singular value decomposition of the vanishing factors in the gradient and random matrices theory, we find that the singular values have to be confined in a narrow interval and derive conditions about their root mean square value. Then, we also improve the efficiency of the training of a recurrent neural network, defining a new method for speeding up this process. Thanks to a least square regularization, we can initialize the parameters of the network, in order to set them closer to the minimum and running fewer epochs of classical training algorithms. Moreover, it is also possible to completely train the network with our initialization method, running more iterations of it without losing in performance with respect to classical training algorithms. Finally, it is also possible to use it as a real-time learning algorithm, adjusting the parameters to the new data through one iteration of our initialization. In the last part of this thesis, we apply recurrent neural networks to non-linear prediction problems. We consider prediction of numerical sequences, estimating the following input choosing it from a probability distribution. We study an automatic text generation problem, where we need to predict the following character in order to compose words and sentences, and a path prediction of walking mobile users in the central area of a city, as a sequence of crossroads. Then, we analyse the prediction of video frames, discovering a wide range of applications related to the prediction of movements. We study the collision problem of bouncing balls, taking into account only the sequence of video frames without any knowledge about the physical characteristics of the problem, and the distribution over days of mobile user in a city and in a whole region. Finally, we address the state-of-the-art problem of missing data imputation, analysing the incomplete spectrogram of audio signals. We restore audio signals with missing time-frequency data, demonstrating via numerical experiments that a performance improvement can be achieved involving recurrent neural networks.

پدیده یخبندان دیررس رخدادی فرین و زیانبخش جوی بوده که در شاخه های مختلف مهندسی مانند کشاورزی، باغبانی و غیره بررسی می شود. تحلیل احتمالی این رخداد و به کارگیری نتایج آن می تواند آسیب های وارده به بخش کشاورزی، باغبانی و غیره را کاهش دهد. موضوع این تحقیق تحلیل آخرین روز یخبندان کلیه ایستگاه های همدید خراسان رضوی است. هشت توزیع احتمالی نرمال، گامبل نوع 1، گامای2 پارامتری، لوگ نرمال 2 و 3 پارامتری، پارتوی تعمیم یافته، حدی تعمیم یافته و پیرسون نوع 3 بر داده ها برازش و پارامترها با هفت روش (5 روش گشتاوری، حداکثر درست نمائی و آنتروپی) برآورد شد. بهترین تابع برازشی برای هر ایستگاه با آزمون K-S انتخاب و آخرین روز یخبندان در دوره های بازگشت مختلف با توزیع برتر پیش بینی شد. بهترین تابع برازشی در ایستگاه ها متفاوت و عمدتا شامل پارتوی تعمیم یافته، پیرسون نوع 3 و لوگ نرمال 3 پارامتری است. بیشترین کاربرد روش برآورد پارامترها مربوط به انواع گشتاورها است. بنابراین نمی توان یک تابع برازشی با یک روش خاص برآورد پارامترها را در بررسی این پدیده پیشنهاد داد. همچنین بررسی روابط رگرسیونی بین دوره های بازگشت مختلف آخرین روز یخبندان و شاخص های دمائی سالانه هر ایستگاه نشان داد که می توان آخرین روز یخبندان را براساس متوسط دمای کمینه هر ایستگاه با دقت مناسب برآورد کرد.

The two-color spotted microarray is a highly effective biotechnology to measure the expression levels of many thousands of genes simultaneously, and it has been commonly used to address various scientific questions in different fields. To date, there are a lot of research works devoted to developing methods of data transformation and data analysis for two-color microarray experiments. However, design issues seem to be relatively unexplored. An appropriate design may guarantee that all the scientific questions of interest are answerable. Also, an efficient design ensures the experimental results as precise as possible subject to scientific and practical constraints. Hence, the statistical design also plays an important role in a two-color microarray study. In the present study, we focus on two major types of experiments in two-color microarray studies: control-test experiments and comparative experiments. We consider the design issues that how to determine which mRNA samples are to be labeled with which fluorescent dye; which mRNA samples are to be hybridized together on the same slide. In the current design framework concerning two-color microarray experiments, random errors are usually assumed to be uncorrelated random variables in the statistical models used to construct designs, but it is often more reasonable to acknowledge that there could be some correlation structure among the random errors. Particularly, when the technical replicates are used in the experiments. The global normalization model proposed by Chai and Liao (2005) is slightly modified to incorporate the possible correlation due to technical replication. Two heuristic algorithms are thus developed to generate A-optimal or highly efficient designs based on the proposed model for the two types of experiments. Fortunately, resulting designs generated from these algorithms are shown to be robust against the correlation due to technique replication. The resulting designs tabularized in the appendix can be immediately used by biologists for conducting two-color microarray experiments. Moreover, some statistical properties and designs with special patterns are also reported.

In this paper, the robust H∞-control problem is reported for a class of uncertain discrete-time fuzzy systems with both multiple probabilistic delays and multiple missing measurements and random missing control from the fuzzy controllers to the actuator. A sequence of random variables including accounting for the probabilistic communication delays and the random missing control are thought as mutually independent and obey the Bernoulli distribution. The measurement-missing phenomenon can be assumed to occur stochastically. Assumption that the missing probability for each sensor satisfies a certain probabilistic distribution in the interval [0 1] is given. Much attention is focused on design of H∞ the fuzzy output feedback controllers to ensure that the resulting close-loop Takagi-Sugeno (T-S) system is exponentially stable in the mean square. The developed method makes disturbance rejection attenuation satisfy a given level by means of the H∞-performance index. Intensive analysis is employed to reach the sufficient conditions about the existence of admissible output feedback controllers which satisfies the exponential stability as well as the prescribed H∞ performance. In addition, the cone-complementarity linearization procedure is utilized to transform the controller-design problem into a sequential minimization one which can be solved by the semi-definite program method. Simulation results conform the feasibility as weil as the effectiveness of the proposed design method.

The first part of this chapter introduces the random walk initially in a form with stochastic shocks generated by a random variable with a binomial distribution. In the simplest version of this process the random variable has two equally likely outcomes resulting in a symmetric binomial random walk. The idea is simple enough and the terminology is due to a problem posed by Pearson (see Hughes, 1995, p.53), although the concept dates from much earlier, originating in games of chance. Starting from the origin, at regular intervals a walker takes equally spaced steps either to the left (north) or to the right (south), with equal probability. The walker’s progress can be plotted in two dimensions by recording the distance from the origin on the vertical axis and the elapsed time on the horizontal axis; for example, one step to the north followed by one step to the south returns the walker to the origin. Such a graph will look like a series of equally sized steps, see Figure 1.1; perhaps surprisingly, the resulting path does not generally oscillate around zero, the theoretical mean of the process. This lack of ‘mean reversion’ is one of the key characteristics of a random walk. In this form the process has its origins in gambling, where the gambler gains or loses an equal amount at each gamble, with equal probability, and intuition might suggest that the gambler is not systematically losing or winning. The ‘distance from the origin’ corresponds to the gambler’s win/lose tally, which is one-dimensional and can be represented on the vertical axis.

Life science students who take undergraduate statistics courses will engage with statistics as consumers, and perhaps even producers, of quantitative research throughout their studies and professions. Several years ago, a study was conducted in an undergraduate statistics course at our institution that was designed to improve the quantitative preparation of life sciences students. Although we observed improvements in students’ confidence in their abilities to choose appropriate statistical procedures and to interpret results from the beginning to the end of the course, there were still noticeable gaps in their abilities to do both at the end of the course. A follow-up survey study was recently conducted in the same course (n=164). This paper will highlight results on students’ self- efficacy and abilities to recognize dependency in data and select appropriate statistical procedures. Implications for future quantitative life science course offerings will also be discussed.

Our institution is heavily invested in improving the learning outcomes of our students, especially in core courses like statistics which is taken by majority of freshmen students. As such, we have been reviewing our existing pedagogy and exploring novel ways to improve students experience in statistics course. In this study we compare web-based vs paper-based traditional homework models and their impact on students’ performance in an undergraduate statistics course. In web-based model, students used MyStatLab to complete the homework assignments along with additional resources offered by this platform. On the other hand, in paper-based model students used traditional approach in completing the homework and was manually graded by the instructor. Our analysis indicated that there is no statistically significant difference in students’ final course grades between the two modalities. Additionally, we conducted a chi-square test of independence and found no statistically significant relationship between students’ final course grades and homework models. However, our analysis concluded that student performance and perceptions improved with online homework model compared to the traditional model. Keywords: Web-based online homework, Traditional homework, Student performance, Undergraduate Statistics DOI: 10.7176/JEP/13-27-09 Publication date: September 30 th 2022

Publication in the conference proceedings of EUSIPCO, Glasgow, Scotland, 2009