In cryptography, periodic sequences with terms in F2 are used almost everywhere. These sequences should have large linear complexity to be cryptographically strong. In fact, the linear complexity of a sequence should be close to its period. In this thesis, we study the expected value for N-periodic sequences with terms in the finite field Fq. This study is entirely devoted to W. Meidl and Harald Niederreiter&rsquo / s paper which is &ldquo / On the Expected Value of the Linear Complexity and the k-Error Linear Complexity of Periodic Sequences&rdquo / We only expand this paper, there is no improvement. In this paper there are important theorems and results about the expected value of linear complexity of periodic sequences.

[pt] Neste trabalho apresentaremos a noção de valor esperado de uma variável aleatória, ou valor médio de uma quantidade aleatória, um conceito probabilístico extremamente importante e útil em diversas aplicações, mas que por razões históricas, não costuma ser ensinado no Ensino Médio. Além desse assunto, abordaremos também alguns problemas interessantes e desafiadores de Probabilidade, como por exemplo, questões dos vestibulares mais difíceis do País, como o do Instituto Militar de Engenharia (IME) e O Desafio em Matemática da PUC-Rio. Em várias das atividades propostas, ao longo nosso trabalho, iremos utilizar recursos computacionais como o Excel e o GeoGebra, e mostrar que podem ser fortes aliados ao ensino de Probabilidade e auxiliar no entendimento do conceito de Valor Esperado. / [en] In this dissertation we present the definition of the expected value of a random variable, an important probabilistic concept which is useful in many applications but which, for historical reasons, is not taught in high school in Brazil. We also discuss examples of interesting and challenging probability problems, including questions from some of the hardest exams in the country, such as the Vestibular for the Instituto Militar de Engenharia (IME) and the Desafio em Matemática of PUC-Rio. In many of the proposed activities, we use computational tools such as Excel and GeoGebra: these can become allies when teaching probability and help in the understanding of the concept of expected value.

Sample average approximation (SAA) is a well-known solution methodology for traditional stochastic programs which are risk neutral in the sense that they consider optimization of expectation functionals. In this thesis we establish sample average approximation methods for two classes of non-traditional stochastic programs. The first class is that of stochastic min-max programs, i.e., min-max problems with expected value objectives, and the second class is that of expected value constrained stochastic programs. We specialize these SAA methods for risk-averse stochastic problems with a bi-criteria objective involving mean and mean absolute deviation, and those with constraints on conditional value-at-risk. For the proposed SAA methods, we prove that the results of the SAA problem converge exponentially fast to their counterparts for the true problem as the sample size increases. We also propose implementation schemes which return not only candidate solutions but also statistical upper and lower bound estimates on the optimal value of the true problem. We apply the proposed methods to solve portfolio selection and supply chain network design problems. Our computational results reflect good performance of the proposed SAA schemes. We also investigate the effect of various types of risk-averse stochastic programming models in controlling risk in these problems.

This thesis presents various models to the fashion industry to predict the profits for some products. To determine the expected performance of each product in 2016, we used tools of game theory to help us identify the expected value. We went further and performed a simple linear regression and used scatter plots to help us predict further the performance of the products of Prada. We used tools of game theory, analytics, and statistics to help us predict the performance of some of Prada's products. We also used the Tableau platform to visualize an overview of the products' performances. All of these tools were used to aid in finding better predictions of Prada's product performances.

„Nun sag, wie hast du’s mit der Messung im Labor?“ So oder ähnlich lautet die sich anhand dieser Arbeit ergebende Gretchenfrage bezüglich biomechanischer Auswertungen und Studien des Laufsports, welche unter Laborbedingungen durchgeführt werden. Hierbei wird angenommen, dass eine Messung im Labor eine valide experimentelle Operationalisierung des Ausdauerlaufens darstellt. Aufgrund der räumlichen Begrenzung kann lediglich eine vergleichsweise geringe Anzahl an einzelnen Versuchen aufgezeichnet werden. Für die statistische Auswertung werden dann zumeist einzelne Parameter der Zeitreihen berechnet, welche wiederum zusammengefasst durch Mittelwerte den Probanden repräsentieren müssen. Bei der Verwendung von diskreten Parametern reduziert sich die aufgenommene Information der Zeitreihe dabei erheblich. Damit einhergehend muss die Frage geklärt werden, ob die Variabilität eines Probanden anhand diskreter Werte oder anhand der gesamten Kurve Beachtung finden muss. Des Weiteren stellt sich die Frage inwieweit das arithmetische Mittel über eine große Anzahl an Versuchen als die den Probanden repräsentierende Kennzahl verwendet und dessen Variabilität mittels einer endlichen Anzahl an Wiederholungen charakterisiert werden kann. Für die Untersuchungen wurden zunächst zwei Studien durchgeführt, wobei die Aufzeichnung von Bodenreaktionskräften und der Winkelgeschwindigkeit bei 100 Läufen an je zwei Messtagen im Labor erfolgte. Die statistischen Auswertungen umfassen sowohl die Betrachtung der Konvergenz von Folgen kumulierter Mittelwerte, Standardabweichungen und Root Mean Square Errors für diskrete Parameter und die gesamten aufgezeichneten Signale der Bodenreaktionskräfte und Winkelgeschwindigkeit als auch die Untersuchung von Prädiktionsbändern. Zudem wurden unterschiedliche Algorithmen zur Bestimmung der minimalen Anzahl an aufzuzeichnenden Versuchen entwickelt. Diese beinhalten nichtlineare Regressionsmodelle für die Anpassung der kumulierten Fläche der Prädiktionsbänder gesamter Kurven und die Analyse der Differenzen aufeinanderfolgender Standardabweichungskurven. Zusammenfassend geht aus dieser Arbeit hervor, dass die postulierte ausreichende und stabile Charakterisierung eines Probanden anhand des arithmetischen Mittels sowie der vollständigen und soliden Beschreibung der Variabilität für diskrete Parameter nicht nachgewiesen werden konnte. Für gesamte Kurven ergab sich ein anderes Bild. Die Probanden konnten anhand der mittleren vertikalen Bodenreaktionskräfte sowie der Bodenreaktionskräfte in anterior-posterior Richtung stabil und ausreichend charakterisiert werden. Für die Bodenreaktionskräfte in mediolateraler Richtung und die Kurve der Winkelgeschwindigkeit wurde dies nicht bestätigt. Die Möglichkeit der Charakterisierung der Variabilität eines Probanden konnte zudem verifiziert werden. Durch Beibehaltung der ursprünglichen Messprozedur ist die Wahrscheinlichkeit sehr hoch, dass der begangene Fehler den Ausgang der statistischen Auswertung beeinflusst und damit Eigenschaften der vorliegenden Grundgesamtheit unter Umständen falsch widerspiegelt. Von einer Verwendung des Mittelwertes diskreter Parameter sollte daher abgesehen werden. Der Fehler sowie dessen unbekanntes Ausmaß sind zum Teil unkontrollierbar und dessen Auswirkungen auf weitere biomechanische Kenngrößen nicht überprüfbar. Die Annahme, dass eine Labormessung als valide experimentelle Operationalisierung des Ausdauerlaufens angesehen werden kann, ist damit hinfällig. Es ist zukünftig notwendig, die Erforschung neuer Aufnahme- und Auswerteprozeduren, die alternative Verwendung gesamter Kurven und die Entwicklung neuer Testverfahren zu forcieren.

Multilevel Random Pulse Width Modulation (RPWM) schemes have drawn increasing attention in the past few years. Multilevel topologies provide high voltage and high power capabilities and random PWM schemes offer reduction in discrete harmonics spectral. This dissertation provides a generalized theory and analysis methods of the standard five-level Weighted RPWM (WRPWM). Equations have been derived to analyze the spectral performance and average switching frequency of the WRPWM output waveform using statistical approach. A modified WRPWM scheme has been proposed. The modified WRPWM scheme is then analyzed with the equations derived from the same approach. The analyzed theoretical spectrum of the standard five-level WRPWM is then compared with the three-level WRPWM scheme and the conventional carrier based PWM scheme. A scaled laboratory prototype diode clamping five-level inverter has been built for verification of the standard and the proposed modified WRPWM schemes. It can be seen that the experimental measurements and the theoretical analyzed results are all in good agreement. Results show the two five-level WRPWM schemes offers significant improvements on the spectrum content than the conventional carrier based PWM scheme. It was found that the five-level WRPWM schemes have successfully suppress the magnitude of third harmonic below 5% of the magnitude of fundamental component and even less for the higher order harmonic components. Research contributions made by the dissertation are: - The proposed modified multilevel WRPWM scheme which utilizing the switching decision redundancy of multilevel inverter to manipulate the harmonic content of the output signal. - The derived mathematical equations of the standard and modified five-level WRPWM scheme for analytical purposes. / Dissertation (MEng (Electrical Engineering))--University of Pretoria, 2005. / Electrical, Electronic and Computer Engineering / unrestricted

In traditional investment decision making, one tool commonly used is the mean variance model, also known as an expected-value variance (EV) model, which evaluates the anticipated payout of different assets with respect to uncertainty where portfolios with higher risk demand higher expected returns from an individual. This thesis adapts this framework to a cost setting where decision makers are evaluating alternative physical assets that carry lifetime cost uncertainty for maintenance. Specifically, this paper examines homeowner choices for their home plumbing systems in the event of a pinhole leak, a tiny pin-sized hole that forms in copper, drinking-water pipes. These leaks can cause substantial damage and cost homeowners thousands of dollars in repairs. Since pinhole leaks are not related to the age of pipe material, a homeowner is subject to the risk of additional costs if a pinhole leak occurs again despite their repair efforts. The EV cost model in this paper defines two discrete choices for the homeowner in the event of a leak; to apply a simple repair at lower cost and higher future cost uncertainty, or to replace their plumbing with new pipe material, usually made of plastic, at a higher upfront cost but lower likelihood of future expenses. The risk preference of homeowners are demonstrated by their repair strategy selection, as well as the level of cost they incur to reduce uncertainty. Risk neutral individuals will select the repair strategy with the lowest lifetime expected cost and high variance, while risk averse homeowners will prefer to replace their plumbing with higher cost but lower variance. Risk averse individuals are also exposed to indirect costs, which is an additional unobserved cost in the form of a risk premium the homeowner is willing to pay to remove all uncertainty of future pinhole leak expense. Expected costs and variances are also higher for regions in the U.S. that experience elevated leak incident rates, known as hotspots. Using this mean variance cost framework, indirect cost can be quantified for homeowners in hotspot regions and compared to the rest of the U.S. to evaluate the magnitude of pinhole leak risk. The EV cost model estimates risk premiums on pinhole leaks to be $442 for homeowners in hotspots and $305 for those in the rest of the U.S. Finally, this paper examines the impact of pinhole leak cost uncertainty on the U.S. economy. Of an estimated $692 million in annual pinhole leak costs to homeowners, this study estimates a lower bound cost of $54 million per year (7.8% of estimated national annual cost) in risk premium that homeowners would be willing to pay to avoid pinhole leak cost uncertainty. Information in this study on the role of risk in home plumbing decisions and indirect costs would be helpful to policymakers and water utility managers as they deal with infrastructure management decisions. Furthermore, the EV cost methodology established in this paper demonstrates an effective use of mean variance modeling under cost uncertainty. / Master of Science / This paper examines homeowner choices for their home plumbing systems in the event of a pinhole leak, a tiny pin-sized hole that forms in copper, drinking-water pipes. These leaks can cause substantial damage and cost homeowners thousands of dollars in repairs. Since pinhole leaks are not related to the age of pipe material, a homeowner is subject to the risk of additional costs if a pinhole leak occurs again despite their repair efforts. This paper also examined costs in regions of the U.S. that experience elevated leak incident rates, known as hotspots. There were two primary choices assessed in this study for homeowners facing pinhole leaks: to either apply a simple repair today at lower cost but take on a higher chance of more pinhole leaks; or to replace their plumbing with new pipe material, usually made of plastic, at a higher overall cost but lower risk of another leak. Using a cost focused investment analysis, it was estimated that homeowners selecting the ‘safer’ replacement strategy would be willing to pay a minimum of $305 in additional cost if able to eliminate all possibility of another leak compared to those who opted for the more ‘riskier’ repair choice. Additionally, homeowners who live in hotspot regions who selected the replacement strategy were estimated to be willing to pay a minimum of $442 in additional cost to avoid pinhole leaks. At a national level, these pinhole leak-avoiding premiums equate to $54 million, about 7.8% of the estimated $692 million in costs spent on fixing pinhole leaks by U.S. homeowners each year. Information in this study on homeowner preferences and pinhole leak would be helpful to policymakers and water utility managers as they deal with infrastructure management decisions.

Multiobjective Stochastic Linear Programming is a relevant topic. As a matter of fact, many real life problems ranging from portfolio selection to water resource management may be cast into this framework. There are severe limitations in objectivity in this field due to the simultaneous presence of randomness and conflicting goals. In such a turbulent environment, the mainstay of rational choice does not hold and it is virtually impossible to provide a truly scientific foundation for an optimal decision. In this thesis, we resort to the bounded rationality and chance-constrained principles to define satisficing solutions for Multiobjective Stochastic Linear Programming problems. These solutions are then characterized for the cases of normal, exponential, chi-squared and gamma distributions. Ways for singling out such solutions are discussed and numerical examples provided for the sake of illustration. Extension to the case of fuzzy random coefficients is also carried out. / Decision Sciences

Multiobjective Stochastic Linear Programming is a relevant topic. As a matter of fact, many real life problems ranging from portfolio selection to water resource management may be cast into this framework. There are severe limitations in objectivity in this field due to the simultaneous presence of randomness and conflicting goals. In such a turbulent environment, the mainstay of rational choice does not hold and it is virtually impossible to provide a truly scientific foundation for an optimal decision. In this thesis, we resort to the bounded rationality and chance-constrained principles to define satisficing solutions for Multiobjective Stochastic Linear Programming problems. These solutions are then characterized for the cases of normal, exponential, chi-squared and gamma distributions. Ways for singling out such solutions are discussed and numerical examples provided for the sake of illustration. Extension to the case of fuzzy random coefficients is also carried out. / Decision Sciences

The purpose with this study is to show how the subjective uncertainty analysis can be applied to the cost estimates and examine how the attitudes towards risk budgeting of construction projects works. Biogas Mellannorrland is based on a case study and focuses on four location options for the biogas plant. For each option, a cost estimate in form of a fixed cost and variable cost elements, where the latter are uncertain. Pearson-Tukey method have been applied as a method to estimating subjective probabilities. An assumption is made so that the uncertain costs are normally distributed, where the parameters’ mean and standard deviation collected from respondents indicating percentiles at 5% and 95% for doubtful cost elements. In order to identify the project managers risk attitude, the risk premium method will be used to calculate where the expected value is produced by Pearson-Tukey method. The expected value is subtracted with certainty equivalent, which consist of the budgeted amount. The study was conducted through interviews in order to obtain the costs and the subjective probabilities. The study results show that it is difficult to determine which option is the best, or the worst, by just looking at the expected value. The risk premium shows that each option is a risk averse choice for the project managers, since the risk premium was greater than zero. / Denna studie syftar till att visa hur subjektiv osäkerhetsanalys kan tillämpas vid kostnadskalkyler samt undersöka riskattityder vid budgetering av byggprojekt. Undersökningen är en fallstudie vid Biogas Mellannorrland och fokuserar på fyra lokaliseringsalternativ för biogasanläggning. För varje alternativ görs en kostnadskalkyl i form av ett antal fasta och rörliga delkostnader, där de senare är osäkra. Som metod för att skatta subjektiva sannolikheter har Pearson-Tukey-metoden tillämpats. Ett antagande görs att osäkra kostnader är normalfördelade, där parametrarna väntevärde och standardavvikelse hämtas från respondenter som anger percentilerna vid 5 % och 95 % för osäkra delkostnader. För att identifiera projektledarnas riskattityd, kommer riskpremium att beräknas, där det förväntade värdet tas fram med Pearson-Tukey- metoden. Det förväntade värdet subtraheras med säkerhetsekvivalensen, vilket utgörs av det budgeterade beloppet. Studien har genomförts med hjälp av intervjuer för att få fram kostnader och subjektiva sannolikheter. Studiens resultat visar att det är svårt att avgöra vilka alternativ som är bäst eller sämst genom att bara titta på väntevärdet. Med riskpremium visar alla alternativ att projektledarna gör ett riskaversivt val, eftersom riskpremium blev större än noll.