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Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods, discretization), PDEs (diffusion equations, scattering theory), representation theory; iterated function systems (fractals, Julia sets, chaotic dynamical systems), ergodic theory, operator algebras, and many more. And neighboring areas, probability/statistics (for example stochastic processes, Ito and Malliavin calculus), physics (representation of Lie groups, quantum field theory), and spectral theory for Schr\"odinger operators. We have strived for a more accessible book, and yet aimed squarely at applications; -- we have been serious about motivation: Rather than beginning with the four big theorems in Functional Analysis, our point of departure is an initial choice of topics from applications. And we have aimed for flexibility of use; acknowledging that students and instructors will invariably have a host of diverse goals in teaching beginning analysis courses. And students come to the course with a varied background. Indeed, over the years we found that students have come to the Functional Analysis sequence from other and different areas of math, and even from other departments; and so we have presented the material in a way that minimizes the need for prerequisites. We also found that well motivated students are easily able to fill in what is needed from measure theory, or from a facility with the four big theorems of Functional Analysis. And we found that the approach "learn-by-using" has a comparative advantage.

Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods, discretization), PDEs (diffusion equations, scattering theory), representation theory; iterated function systems (fractals, Julia sets, chaotic dynamical systems), ergodic theory, operator algebras, and many more. And neighboring areas, probability/statistics (for example stochastic processes, Ito and Malliavin calculus), physics (representation of Lie groups, quantum field theory), and spectral theory for Schr\"odinger operators. We have strived for a more accessible book, and yet aimed squarely at applications; -- we have been serious about motivation: Rather than beginning with the four big theorems in Functional Analysis, our point of departure is an initial choice of topics from applications. And we have aimed for flexibility of use; acknowledging that students and instructors will invariably have a host of diverse goals in teaching beginning analysis courses. And students come to the course with a varied background. Indeed, over the years we found that students have come to the Functional Analysis sequence from other and different areas of math, and even from other departments; and so we have presented the material in a way that minimizes the need for prerequisites. We also found that well motivated students are easily able to fill in what is needed from measure theory, or from a facility with the four big theorems of Functional Analysis. And we found that the approach "learn-by-using" has a comparative advantage.

DRS 113 Probability Theory Lecture Notes collection at Yangon University of Economics

95 p., 16 p. ; 21 cm

The 1992-1993 year in Control Theory focused on robotics, optimal control for nonsmooth geometric data, adaptive control and filtering, discrete event systems, and power systems. It brought together mathematicians, electrical engineers and practitioners, who identified current difficulties and challenges. The 1993-1994 year in Emerging Applications of Probability brought together computer scientists and probabilists; it also dealt with stochastic models in geosystems, hidden Markov processes in speech and vision, stochastic networks and branching processes. The 1994-1995 year in Waves and Scattering dealt with random waves, numerical methods including hybrid methods (FFT and wavelets), and inverse problems, such as nondestructive evaluation, acoustics, NMR, Electro-magnetic, and ground penetrating radar.

The 1992-1993 year in Control Theory focused on robotics, optimal control for nonsmooth geometric data, adaptive control and filtering, discrete event systems, and power systems. It brought together mathematicians, electrical engineers and practitioners, who identified current difficulties and challenges. The 1993-1994 year in Emerging Applications of Probability brought together computer scientists and probabilists; it also dealt with stochastic models in geosystems, hidden Markov processes in speech and vision, stochastic networks and branching processes. The 1994-1995 year in Waves and Scattering dealt with random waves, numerical methods including hybrid methods (FFT and wavelets), and inverse problems, such as nondestructive evaluation, acoustics, NMR, Electro-magnetic, and ground penetrating radar.

The 1992-1993 year in Control Theory focused on robotics, optimal control for nonsmooth geometric data, adaptive control and filtering, discrete event systems, and power systems. It brought together mathematicians, electrical engineers and practitioners, who identified current difficulties and challenges. The 1993-1994 year in Emerging Applications of Probability brought together computer scientists and probabilists; it also dealt with stochastic models in geosystems, hidden Markov processes in speech and vision, stochastic networks and branching processes. The 1994-1995 year in Waves and Scattering dealt with random waves, numerical methods including hybrid methods (FFT and wavelets), and inverse problems, such as nondestructive evaluation, acoustics, NMR, Electro-magnetic, and ground penetrating radar.

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin

Research partially supported by this grant culminated in the submission of twenty eight new research papers twenty three of which have now been accepted for publication. Topics covered in this research include theory of large deviations, stochastic differential games, stochastic control, stochastic networks, random graphs, mean field models, and diffusion approximations. Four Ph.D. students (Xin Liu, Dominik Reinhold, Jiang Chen and Xuan Wang) completed their dissertation under the direction of the PI. Three other Ph.D. students (Abhishek Pal Majumdar, Ruoyu Wu and Eric Friedlander) continued work on their dissertations. Postdoctoral fellows C.Y.Lee, X.Song, S.Saha and L.Fan were partially supported by this grant. Research was disseminated through numerous invited presentations.

Research partially supported by this grant culminated in the submission of twenty eight new research papers twenty three of which have now been accepted for publication. Topics covered in this research include theory of large deviations, stochastic differential games, stochastic control, stochastic networks, random graphs, mean field models, and diffusion approximations. Four Ph.D. students (Xin Liu, Dominik Reinhold, Jiang Chen and Xuan Wang) completed their dissertation under the direction of the PI. Three other Ph.D. students (Abhishek Pal Majumdar, Ruoyu Wu and Eric Friedlander) continued work on their dissertations. Postdoctoral fellows C.Y.Lee, X.Song, S.Saha and L.Fan were partially supported by this grant. Research was disseminated through numerous invited presentations.

This is the second of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. This paper outlines how the framework can assist in the development of homotopy probability theory, where a vector space of random variables is replaced by a chain complex of random variables. This allows the principles of derived mathematics to participate in classical and noncommutative probability theory. A simple example is presented.

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin, tutorial 12 on 2021-05-09. Erratum: Попытка расчёта интеграла ∫ x dx = 1/2 вероятностными методами (1:56:40) мне не удалась: я описал правильную вероятностную задачу и правильно (хотя и сложновато) получил в ней ответ 1/2, но не показал, почему этот ответ равен ∫  x   dx. Это делается так же, как я потом сделал для интеграла ∫ x n   dx.

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2022, remote tutorials by Alex Shpilkin, tutorial 12 on 2022-05-15.

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2022, remote tutorials by Alex Shpilkin, tutorial 12 on 2022-05-15.

Introduction: Selecting suitable crops for cultivation in a non-certain environment is considered as an important management topic in the agricultural sector. Despite the multiple application of probability theory in quantifying uncertainty in the form of risk programming, validity of this theory depends on the existence of frequency for uncertain variable. For events that cannot be measured by frequency, the only solution is to use subjective judgment of persons in the domain field rather than historical data. Some experts have mistakenly considered subjective judgmentl as a subjective probability and thus used the probability theory to quantify subjective judgmental. But based on existing evidence, the quantification of subjective judgment should be carried out in another theory called the uncertainty theory. In uncertainty theory, in addition to using the belief degree rather than frequency for calculating mathematical moments, the expected value of multiplicative variables will be different with their corresponding relations in the probability theory. Considering these conditions and having in mind that the agricultural sector is always faced with uncertain variables such as price of crops and weather conditions like rainfall, in this study the revenue uncertainty measures of major crops in the Goharbaran region of Sari have been calculated and compared. There are different measures for uncertainty, which in the present study variance and Tail Value at Risk (TVaR) have been used . Materials and Methods: The first step in the application of the uncertainty theory is the elicitation of the belief degree or subjective judgments of the farmers about the crop's price and rainfall during the crop season. To elicit the uncertainty distribution of these variables based on the subjective judgments of farmers, 120 farmers were randomly selected in 2018. After eliciting the farmers' beliefs about uncertain rainfall and prices in the cdf method, it was necessary to select the number of belief degree which current practice was based on previous studies in this field. After calculating the above subjective judgments, while assuming linear, zigzag, normal and normal forms for uncertainty distribution, the parameters of each function were calculated using the least squares method. Among the forms of uncertainty distribution functions, the best form of the uncertainty distribution for each crop's price and rainfall was selected by comparing the RMSE indexes. Subsequently, by calculating a causal relationship between rainfall and crop yield, inverse uncertainty distribution of yield was also extracted. Given the inverse uncertainty distribution functions of crop price and yield, required parameters such as expected revenue, variance and TVaR of revenue at 95% confidence were calculated based on operational laws of uncertainty theory and probability theory. Eviews and Matlab software were used to estimate the yield response function and the uncertainty distribution functions, respectively. Results and Discussion: In this study, after collecting the belief degree of farmers in the studied area about different levels of price and rainfall, three groups of comprehensive beliefs about prices and rainfall were determined by goodness of fit test. Then, according to the relationship between crop yield and rainfall, the inverse function uncertainty distribution is also calculated. With the uncertainty distribution function of crops price and yield, the expected revenue, variance (standard deviation) and TVaR measure for revenue per hectare of crops were calculated and compared with the uncertainty theory as well as probability theory. Based on the results of this study, the amount of the above measures varied in different belief degree groups,     which is due to differences in the uncertainty distribution parameters. Also, based on the results of this study in all groups of beliefs for all crops, the probability theory compared to the uncertainty theory has estimated the variance approximately more than 30% less, which is a significant result. In other words, applying probability theory to belief modeling will lead to erroneous and misleading results. In the case of the TVaR measure in binary multiplicative variable conditions, the use of probability theory and uncertainty theory in calculating TVaR does not yield conflicting results. Conclusion: The purpose of this study was to compare the results of applying probability theory for modeling belief degree rather than uncertainty theory in order to illustrate the necessity of using uncertainty theory in belief degree modeling. Studying the effect of probability theory in modeling the belief degree also suggests that the application of probability theory in the presence of two uncertain variables has no significant effect on expected values and TVaR but has a significant effect on variance size. Based on the results of the present study, assuming the binary multiplicative variable, in calculating higher mathematical moments such as variance, the results of probability theory and uncertainty theory make a considerable difference. This demonstrates the need to promote the uncertainty theory in belief degree modeling. In other words, basic training in the belief degree modeling method should be considered.

Kolmogorov II

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin, tutorial 10 on 2021-04-25.

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin, tutorial 14 on 2021-05-23. Incomplete recording. Addendum: Уравнение 4 t 3 (1− t ) 2 − 2 t 4 (1− t ) = 0 выглядит страшно, но всё становится просто, если заметить общий множитель t 3 (1− t ) и вынести его. Получается t 3 (1− t )(4−4 t −2 t ) = 0, а произведение равно нулю тогда и только тогда, когда равен нулю один из его множителей, так что кандидатами на максимум получаются точки t = 0, t = 1 и t = 4/6. Первые две — это края отрезка [0,1]; в них правдоподобие равно нулю (они соответствуют тому, что монета всегда падает только одной стороной, а по нашим данным получается оба варианта), так что это минимумы, а не максимумы. Поскольку максимум хоть где-нибудь да будет, он должен быть в последней точке. Таким образом, оценка максимального правдоподобия совпала с очевидной оценкой 4 / (4+2) = (число успехов) / (число испытаний). Теперь, когда мы знаем, что делать, нетрудно сделать этот расчёт и в общем виде, для произвольных k и l.  Правдоподобие, как и раньше, есть p k (1− p ) l , так что изменение правдоподобия при небольшом отклонении ε от кандидата t на максимум равно ( t +ε) k [(1− t )−ε] l . Раскрывать скобки в общеми виде было бы грустно, но это и не нужно: как мы только что поняли в случае k = 4, l = 2, нас интересуют только слагаемые нулевой и первой степени по ε, а остальные можно даже не выписывать. [На самом деле и нулевой не нужны, потому что мы заранее знаем, что они в итоге дадут  t k (1− t ) l , но их проще оставить.] Первая скобка по биному Ньютона даёт C k 0 t k + C k 1 t k −1 ε + ⋯ = t k + t k −1 ε + ⋯ , вторая C l 0 (1− t ) l  − C l 1 (1− t ) l −1 ε + ⋯ = (1− t ) l + (1− t ) l −1 ε + ⋯ (если знаете производную степенной функции — заметьте, что это она), их произведение равно  t k (1− t ) l  + [ kt k −1 (1− t ) l  − lt k (1− t ) l −1 ]ε + ⋯ , так что остаётся, как и раньше, приравнять нулю коэффициент  kt k −1 (1− t ) l  −  lt k (1− t ) l −1  при ε (производная произведения!) и решить получившееся уравнение (случаи k = 0 и l = 0, когда максимум оказывается на краю отрезка, в принципе надо разобрать отдельно, но ответ в них совпадает с полученным по формуле ниже). Снова вынося общий множитель, получаем уравнение  t k −1 (1− t ) l −1 [ k (1− t ) −  l ] и корни t = 0 (если k > 1), t = 1 (если l > 1) и t = k / ( k + l ). При k , l ≠ 0 максимум на краю отрезка невозможен по той же причине, что и раньше, так что оценка максимального правдоподобия есть  k  / ( k + l ) и опять совпадает с очевидной (число успехов) / (число испытаний).

This study replicates Lagattuta, Sayfan, and Harvey (2014) in an online experiment with ~70 adults recruted on prolific. In this task, each trial begins with a simple picture (a cloud, a castle, or a panda) that is then covered so only a small 1.5"×1.5" fragment remains visible. Informed participants first view and label the full picture, then see only the fragment. Uninformed participants see only the fragment from the start. All participants are asked to judge what a naïve observer—who has never seen the full picture—would think is behind the cover. They first freely guess, then rate a set of seven possible images (the true picture, a basic shape, and various more or less plausible alternatives) on a 0–10 probability scale, removing the ones they believe the observer "definitely would not" choose. By comparing ratings from those who knew the whole image versus those who did not, the task reveals how privileged knowledge distorts our predictions of another person’s thoughts: informed participants overestimate the chance that the observer would pick the correct picture and downplay options that seem plausible when the fragment is viewed in isolation.

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin, tutorial 11 on 2021-05-02. Incomplete recording. Addendum: Все эти (пока безрезультатные) рассуждения про интегралы были нацелены на то, чтобы как-то обосновать теорему де Муавра—Лапласа про приближение биномиального распределения нормальным. Нам понадобятся-таки значения чисел в этом нормальном распределении; целиком оно выглядит как 1/√(2πσ 2 ) × e i 2 /2σ 2 , где σ 2 , которое при внимательном рассмотрении оказывается дисперсией этого распределения, равно дисперсии исходного биномиального, то есть 300. Следующий шаг заключается в том, чтобы приблизить сумму значений уже этого нормального распределения в целых точках от − k до k его интегралом от − k до k , как мы до этого делали для логарифма. Наконец, нужно узнать, при каком k этот интеграл надо обрезать, чтобы результат получился равен 99% (в нашей задаче) или больше. Никакой полезной формулы тут написать нельзя, но люди обычно помнят стандартный набор значений: от −σ до σ лежит ≳ 68% (примерно треть в хвостах), от −2σ до 2σ лежит ≳ 95% (примерно одна двадцатая в хвостах), а от −3σ до 3σ лежит ≳ 99,7% (несколько тысячных в хвостах). Остальные значения можно посмотреть в специальных (полученных численно) таблицах; например, нужный нам однопроцентный хвост нужно отрезать примерно на 2,5 σ. Оба приближения, которые мы использовали в этом процессе (биномиального распределения нормальным и суммы интегралом) для симметричного распределения с параметром 300 (как у нас) имеют пренебрежимо малую ошибку; они вполне прилично работают даже с параметром 10. В итоге в задаче про гардеробы получается так: если положить σ 2 = 300, то там, где неравенство Чебышёва предписывало нам отрезать хвост на 10σ, приближение нормальным распределением говорит, что его достаточно отрезать на 3σ или даже меньше, и (тем же способом) вместо 250 с лишним вешалок с каждой стороны их получается нужно чуть меньше 180. Тут правильно отрезать хвосты с обоих сторон: непопадание в правый хвост означает, что правый гардероб не переполнен, а непопадание в левый — что левый, и оба условия нам нужны. Erratum: В действительности правильное приближение в симметричных координатах выглядит как n ! / 2 n [( n − k )/2]![( n + k )/2]! ≈ 2/√(2π n ) ×  e k 2 /2 n , с лишней с виду двойкой в числовом коэффициенте спереди. Его можно получить, либо честно перекосив координаты в стандартной формулировке теоремы де Муавра — Лапласа, либо из менее вычислительных соображений, которые мы ещё обсудим.

Book Source: Digital Library of India Item 2015.234003 dc.contributor.author: Reichenbach Hans dc.date.accessioned: 2015-07-10T18:59:37Z dc.date.available: 2015-07-10T18:59:37Z dc.date.digitalpublicationdate: 2005-12-23 dc.date.citation: 1949 dc.identifier.barcode: 99999990131322 dc.identifier.origpath: /rawdataupload1/upload/0129/984 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/234003 dc.description.scanningcentre: C-DAC, Noida dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 510 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: University Of California Press, Berkeley And Los Angeles dc.rights: Regents Of The University Of California dc.source.library: Ratan Tata Library, University Delhi dc.subject.classification: Natural Sciences dc.title: The Theory Of Probability

Recently, a manner of articles have appeared in newspapers and popular magazines and journals emphasizing a dichotomy in regard to fuzzy sets: the great potential benefit of using an intuitively appealing and simple approach to modeling uncertainties and its almost universal rejection or avoidance by orthodox scientists in the United States trained in probability theory. Thus, once more, it is of some interest to attempt to determine dispassionately what relations, if any, exist between fuzzy sets and probabilities. Previously, Goodman et al. were the first to point out that rather basic connections do indeed exist between fuzzy set theory and probability theory via random sets and their one point coverage functions. (See, e.g. Goodman, Some new results concerning random sets and fuzzy sets, info. Sci. 34, 1984 or Goodman & Nguyen, Uncertainty Models for Knowledge-Based Systems (monograph), North-Holland Co., Amsterdam, 1985.) But relatively few individuals have utilized these connections (Dubois & Prade on occasion and Oblow emphasizing full random set representations through his hybrid O-Theory), mainly due to the foreboding complex structure of random sets - as compared to the relatively simple form of random variables or random vectors. Even the relatively recent contributions of Lindley, Klir, and others in comparing the roles and game theoretic admissibility properties of fuzzy sets and probabilities do not address the deeper relations between the two areas.

Recently, a manner of articles have appeared in newspapers and popular magazines and journals emphasizing a dichotomy in regard to fuzzy sets: the great potential benefit of using an intuitively appealing and simple approach to modeling uncertainties and its almost universal rejection or avoidance by orthodox scientists in the United States trained in probability theory. Thus, once more, it is of some interest to attempt to determine dispassionately what relations, if any, exist between fuzzy sets and probabilities. Previously, Goodman et al. were the first to point out that rather basic connections do indeed exist between fuzzy set theory and probability theory via random sets and their one point coverage functions. (See, e.g. Goodman, Some new results concerning random sets and fuzzy sets, info. Sci. 34, 1984 or Goodman & Nguyen, Uncertainty Models for Knowledge-Based Systems (monograph), North-Holland Co., Amsterdam, 1985.) But relatively few individuals have utilized these connections (Dubois & Prade on occasion and Oblow emphasizing full random set representations through his hybrid O-Theory), mainly due to the foreboding complex structure of random sets - as compared to the relatively simple form of random variables or random vectors. Even the relatively recent contributions of Lindley, Klir, and others in comparing the roles and game theoretic admissibility properties of fuzzy sets and probabilities do not address the deeper relations between the two areas.

Recently, a manner of articles have appeared in newspapers and popular magazines and journals emphasizing a dichotomy in regard to fuzzy sets: the great potential benefit of using an intuitively appealing and simple approach to modeling uncertainties and its almost universal rejection or avoidance by orthodox scientists in the United States trained in probability theory. Thus, once more, it is of some interest to attempt to determine dispassionately what relations, if any, exist between fuzzy sets and probabilities. Previously, Goodman et al. were the first to point out that rather basic connections do indeed exist between fuzzy set theory and probability theory via random sets and their one point coverage functions. (See, e.g. Goodman, Some new results concerning random sets and fuzzy sets, info. Sci. 34, 1984 or Goodman & Nguyen, Uncertainty Models for Knowledge-Based Systems (monograph), North-Holland Co., Amsterdam, 1985.) But relatively few individuals have utilized these connections (Dubois & Prade on occasion and Oblow emphasizing full random set representations through his hybrid O-Theory), mainly due to the foreboding complex structure of random sets - as compared to the relatively simple form of random variables or random vectors. Even the relatively recent contributions of Lindley, Klir, and others in comparing the roles and game theoretic admissibility properties of fuzzy sets and probabilities do not address the deeper relations between the two areas.

Book digitized by Google from the library of the New York Public Library and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the New York Public Library and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

2nd ed., re-written and greatly enlarged

2nd ed., re-written and greatly enlarged

2nd ed., re-written and greatly enlarged

The definition of the variance of f is equivalent to its interpretation as the mean square of f's deviation from its mean. However, it can be used to put an upper limit on the probability that a random variable takes on values with deviations greater than x� for any x greater than 1. The simplest and worst such bound is called Tchebyshev's inequality.

We elaborate on the proposed general boundary formulation as an extension of standard quantum mechanics to arbitrary (or no) backgrounds. Temporal transition amplitudes are generalized to amplitudes for arbitrary spacetime regions. State spaces are associated to general (not necessarily spacelike) hypersurfaces. We give a detailed foundational exposition of this approach, including its probability interpretation and a list of core axioms. We explain how standard quantum mechanics arises as a special case. We include a discussion of probability conservation and unitarity, showing how these concepts are generalized in the present framework. We formulate vacuum axioms and incorporate spacetime symmetries into the framework. We show how the Schroedinger-Feynman approach is a suitable starting point for casting quantum field theories into the general boundary form. We discuss the role of operators.

We establish connections between the requirement of measurability of a probability space and the principle of complimentarity in quantum mechanics. It is shown that measurability of a probability space implies the dependence of results of quantum measurement not only on the properties of a quantum object under consideration, but also on the classical characteristics of the measuring device which is used. We show that if one takes into account the requirement of measurability in a quantum case, the Bell inequality does not follow from the hypothesis about the existence of an objective reality.

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin, tutorial 13 on 2021-05-16. Incomplete recording.

Estimating the probability that a proposed bid will win is obviously a central problem in competitive bidding theory. Unfortunately, a method for doing this, which originated with Friedman and which continues to be expounded in the literature, is logically incorrect. This paper points out the deficiency in that formulation. (Author/MLF)

Probability Theory and Related Fields 1986: Volume 71 , Issue Contents. Digitized from IA1631302-05 . Next issue: sim_probability-theory-and-related-fields_1986-01_71_1 .

Probability Theory and Related Fields 1986: Volume 71 , Issue Contents. Digitized from IA1631302-05 . Next issue: sim_probability-theory-and-related-fields_1986-01_71_1 .

Probability Theory and Related Fields 1986: Volume 71 , Issue Contents. Digitized from IA1631302-05 . Next issue: sim_probability-theory-and-related-fields_1986-01_71_1 .

Probability Theory and Related Fields 1986: Volume 71 , Issue Contents. Digitized from IA1631302-05 . Next issue: sim_probability-theory-and-related-fields_1986-01_71_1 .

Probability Theory and Related Fields 1986: Volume 71 , Issue Contents. Digitized from IA1631302-05 . Next issue: sim_probability-theory-and-related-fields_1986-01_71_1 .

Measure Theory Probability ( 1)

This article is from Frontiers in Psychology , volume 5 . Abstract None

Probability Theory and Related Fields 1987: Volume 76 , Issue Contents. Digitized from IA1631302-05 . Previous issue: sim_probability-theory-and-related-fields_1987-08_75_4 . Next issue: sim_probability-theory-and-related-fields_1987-09_76_1 .

Probability Theory and Related Fields 1993: Volume 97 , Issue Contents. Digitized from IA1631302-05 . Previous issue: sim_probability-theory-and-related-fields_1993-09_96_4 . Next issue: sim_probability-theory-and-related-fields_1993-10_97_1-2 .