Runway friction testing is performed in order to enhance the safety of aircraft operation on runways. Preventative maintenance friction surveys are performed to determine if there is any deterioration of the frictional resistance on the surface over a period of time and to determine if there is a need for corrective maintenance. In addition operational performance friction surveys are performed to determine frictional properties of a pavement surface in order to provide corrective action information in maintaining safe take-off or landing performance limits. A major issue encountered in both types of friction evaluation on runways is the standardization of the friction measurements from different Continuous Friction Measuring Equipment (CFME). The International Friction Index (IFI) has been formulated to address the above issue and determine the friction condition of a given runway is a standardized format. The ASTM recommended standard procedure to compute the IFI of a runway surface employs two distinct parameters to express the IFI; F60 is the friction value adjusted to a slip speed of 60 km/h and correlated to the standard Dynamic Friction Tester (DFT) measurement. And Sp is the speed constant which is governed by the mean profile depth of that surface. The primary objective of this thesis is to investigate the reliability of the current ASTM procedure to standardize runway friction measurements in terms of IFI. Based on the ASTM standard procedure, two equipment specific calibration constants (A and B) are assigned for each CFME during calibration. Then, in subsequent testing those calibrations constants can be used to adjust the equipment measurements to reliable IFI values. Just as much as A and B are presumed to be characteristic of any given CFME, they are also expected to be independent of the operational speed. The main objective of the annual NASA Runway Friction Workshop held in Wallops Island, Virginia, is to calibrate commonly used CFMEs such that all calibrated equipment would provide a standard reading (i.e. IFI) on a particular surface. During validation of the existing ASTM procedure using the NASA Runway Friction Workshop data it was observed that the single value-based IFI predictions of the calibrated CFMEs were inaccurate resulting in low correlations with DFT measured values. Therefore, a landing pilot should not be left to make a safe decision with such an uncertain single standard friction value because the actual standard friction value could very well be much less than this value. Hence a modified procedure was formulated to treat the calibration constants A and B as normally distributed random variables even for the same CFME. The new procedure can be used to predict the IFI (F60) of a given runway surface within a desired confidence interval. Since the modified procedure predicts a range of IFI for a given runway surface within two bounds, a landing pilot's decision would be made easier based on his/her experience on critical IFI values. However, even the validation of the modified procedure presented some difficulties since the DFT measurements on a few validated surfaces plotted completely outside the range of F60 predicted by the modified method. Furthermore, although the ASTM standard stipulates the IFI (F60) predictions to be independent of the testing speed, data from the NASA Runway Friction Workshop indicates a significant difference in the predictions from the two testing speeds of 65 km/hr and 95 km/hr, with the results from the 65 km/hr tests yielding better correlations with the corresponding DFT measurements. The above anomaly could be attributed to the significantly different FR60 values obtained when the 65 km/hr data (FR65) and 95 km/hr data (FR95) are adjusted to a slip speed of 60 km/hr. Extended analytical investigations revealed that the expected testing speed independency of the FR60 for a particular CFME cannot be supported by the ASTM defined general linear relationship between Sp and the mean profile depth which probably has been formulated to satisfy a multitude of CFMEs operating on a number of selected test surfaces. This very reason can also be attributed to the above mentioned outliers observed during the validation of the modified procedure.

This paper presents a simple recursive formula to compute the Taylor polynomial of quantile for a continuous random variable. It is very easy to implement the formula in standard symbolic programming system, for example Mathematica (Wolfram, 2003). Applications of the formula to standard normal distribution and to the generation of random variables for continuous distribution with bounded support are illustrated.

No description available.

The main objective of this work will be to examine the hypothesis that all the treatment means are the same and equal to some unknown quantity, when we know that the variance is the same for each sample, and to determine if the conventional method for making this test (the F-test) applicable when the sample sizes are assumed to be random variables. This hypothesis can be tested by using a likelihood-ration test. To do this, a density function or distribution has to be found for this ratio, thus permitting us to make probability statements about the occurrence of this ration under the null hypothesis.

ABSTRACT The network of pipes in potable water distribution systems (WDS) are comprised of thousands of pipes made of various materials including PVC, concrete, cast iron, and steel, among several others. The pipes are subjected to internal and external conditions that lead to their failure. Stress conditions include, but are not limited to internal pressures, traffic loading, and corrosion. The deterioration of a pipe decreases its mechanical strength which results in an increase of its probability of failure. Failures lead to loss of service which translates to loss of money due to the cost of repairs and buildup of traffic caused by street closures. The focus of this study is the pipe network underneath cities that make it possible for communities to have access to potable water. The objective of this analysis is to evaluate the physical conditions of each pipe in a water distribution system in order to assess its probability of failure and ultimately calculate the risk associated with each pipe in the case that it were to fail. This model focuses only on the pipes of the WDS and does not take into consideration fittings, pumps, and other network components. This model assesses pipe age, material, diameter, internal pressure, traffic loading (industrial or residential), and length to determine the probability of failure. It then utilizes several economic factors such as material cost, customer criticality, demand, traffic impact, and land use to calculate the risk associated with each pipe. The risk associated with each pipe can then be used as a ranking system to identify the most vulnerable pipes, those with the highest economic impact upon failure. Identifying the pipes with the highest risk allows municipalities to better allocate funds for maintenance or replacement of pipes. It highlights the most critical pipes within a network of thousands. In order to check its functionality, this model applied to the WDS of the City of Arroyo Grande, California. Information on the City’s distribution system was analyzed using Bentley’s WaterCAD, ESRI’s ArcGIS, MathWorks’ MATLAB and Microsoft’s Excel software to perform the analysis. The risk analysis model provided 3 pipes within the distribution system made of cast iron as having a high probability of failure and a critical level of risk. A critical level of risk is defined as falling within the highest range of risk within this study. Considering that only 3 pipe segments were highlighted as having a Critical Risk, 4 as High Risk, and 6 as Medium Risk, in a system of 3572 pipes indicates that the model functions properly. This model was compared to a method developed by Jan C. Devera in his thesis “Risk Assessment Model for Pipe Rehabilitation and Replacement in a Water Distribution System” (2013), which was also applied to the City of Arroyo Grande’s distribution system. Results provided by this analysis prove that both models are functional due to similar results. The current study utilizes the concepts of random variables and conditional assessment to run various Monte Carlo Simulations as the means of calculating the probability of failure of a pipe. Mr. Devera’s model utilizes simplistic approach that does not involve intensive calculations, but results for both models turned out to be similar when looking at the Arroyo Grande distribution system. This risk assessment model demonstrates that a risk assessment model can provide a framework to prioritize pipes based on risk. The approach can help create a schedule for a city’s pipe distribution network for maintenance and repair. It is important to note that it is not a predictive model. This study may be employed to better allocate funds for the rehabilitation and replacement of a city’s existing pipe network to promote optimal operating conditions and service to the public.

The handling of variability effects in structural models is a natural and necessary extension of deterministic analysis techniques. In the context of finite element and uncertainty modelling, the stochastic finite element method (SFEM), grouping the perturbation SFEM, the spectral SFEM and the Monte-Carlo simulation, has by far received the major attention. <br> The present work focuses on second moment approaches, in which the first two statistical moments of the structural response are estimated. Due to its efficiency for handling problems involving low variability levels, the perturbation method is selected for characterising the propagation of the parameter variability from an uncertain dynamic model to its structural response. A dynamic model excited by a time-harmonic loading is postulated and the extension of the perturbation SFEM to the frequency domain is provided. This method complements the deterministic analysis by a sensitivity analysis of the system response with respect to a finite set of random parameters and a response surface in terms of a Taylor series expansion truncated to the first or second order is built. Taking into account the second moment statistical data of the random design properties, the response sensitivities are appropriately condensed in order to obtain an estimation of the response mean value and covariance structure. <br> In order to handle a wide definition of variability, a computational tool is made available that is able to deal with material variability sources (material random variables and fields) as well as shape uncertainty sources. This second case requires an appropriate shape parameterisation and a shape design sensitivity analysis. The computational requirements of the tool are studied and optimised, by reducing the size of the random dimension of the problem and by improving the performances of the underlying deterministic analyses. In this context, modal approaches, which are known to provide efficient alternatives to direct approaches in frequency domain analyses, are developed. An efficient hybrid procedure, coupling the perturbation and the Monte-Carlo simulation SFEM, is proposed and analysed. <br> Finally, the developed methods are validated, by resorting mainly to the Monte-Carlo simulation technique, on different numerical applications: a cantilever beam structure, a plate bending problem (involving a 3-dimensional model), an articulated truss structure and a problem involving a plate with a random flatness default. The propagation of the model uncertainty in the response FRFs and the effects involved by random field modelling are examined. Some remarks are stated pertaining to the influence of the parameter PDF in simulation-based methods. <br> <br> La gestion de la variabilité présente dans les modèles structuraux est une extension naturelle et nécessaire des techniques de calcul déterministes. En incorporant la modélisation de l'incertitude dans le calcul aux éléments finis, la méthode des éléments finis stochastiques (groupant l'approche perturbative, l'approche spectrale et la technique de simulation Monte-Carlo) a reçu une large attention de la littérature scientifique. <br> Ce travail est orienté sur les approches dites de second moment, dans lesquelles les deux premiers moments statistiques de la réponse de la structure sont estimés. De par son aptitude à traiter des problèmes caractérisés par de faibles niveaux de variabilité, la méthode perturbative est choisie pour propager la variabilité des paramètres d'un modèle dynamique incertain sur sa réponse. Un modèle sous chargement dynamique harmonique est supposé et l'extension dans le domaine fréquentiel de l'approche perturbative est établie. Cette méthode complète l'analyse déterministe par une analyse de sensibilité de la réponse du système par rapport à un ensemble fini de variables aléatoires. Une surface de réponse en termes d'un développement de Taylor tronqué au premier ou second ordre peut alors être écrit. Les sensibilités de la réponse sont enfin condensées, en tenant compte des propriétés statistiques des paramètres de design aléatoires, pour obtenir une estimation de la valeur moyenne et de la structure de covariance de la réponse. <br> Un outil de calcul est développé avec la capacité de gestion d'une définition large de la variabilité: sources de variabilité matérielle (variables et champs aléatoires) ainsi que géométrique. Cette dernière source requiert une paramétrisation adéquate de la géométrie ainsi qu'une analyse de sensibilité à des paramètres de forme. Les exigences calcul de cet outil sont étudiées et optimisées, en réduisant la dimension aléatoire du problème et en améliorant les performances des analyses déterministes sous-jacentes. Dans ce contexte, des approches modales, fournissant une alternative efficace aux approches directes dans le domaine fréquentiel, sont dérivées. Une procédure hybride couplant la méthode perturbative et la technique de simulation Monte-Carlo est proposée et analysée. <br> Finalement, les méthodes étudiées sont validées, principalement sur base de résultats de simulations Monte-Carlo. Ces résultats sont relatifs à plusieurs applications numériques: une structure poutre-console, un problème de flexion de plaque (modèle tridimensionnel), une structure en treillis articulé et un problème de plaque présentant un défaut de planéité aléatoire. La propagation de l'incertitude du modèle dans les fonctions de réponse fréquentielle ainsi que les effets propres à la modélisation par champs aléatoires sont examinés. Quelques remarques relatives à l'influence de la loi de distribution des paramètres dans les méthodes de simulation sont évoquées.

Šiame darbe nagrinėjama atsitiktinio dėmenų (atsitiktinių dydžių) skaičiaus skirstinio funkcijos aproksimacijos normaliuoju dėsniu tolygusis ir netolygusis įverčiai. Atsitiktinių dydžių atsitiktinio dėmenų skaičiaus suma traktuojama kaip draudėjo suminės išmokos. Pilnai išnagrinėtas atvejis, kai atsitiktinis dėmenų skaičius yra homogeninis Puasono procesas. Kita darbo dalis skirta atvejui, kai nėra Puasono procesas. Darbą sudaro 8 dalys: įvadas, analitinė – metodinė dalis, rezultatai, kuriais remsimės darbe, darbo tikslas, darbo rezultatai, publikuotas straipsnis, išvados, literatūros sąrašas. Darbo apimtis – 49 p. teksto be priedų, 2 iliustr., 4 bibliografiniai šaltiniai. / This work analyses the even and not even evaluations of distribution function approximation of random amount of random variables using normal law. The sum of random amount of random variables is treated as assurers total payout. In this paper there is fully explored the case when (random amount of random values) is homogeneous Poisson process. The other part of this work explores the case when is not a Poisson process. Structure: introduction, analytical – methodical part, results, are used in work, woprk point, work results, conclusions, references. Thesis consist of: 49 p. text without appendixes, 2 pictures, 4 bibliographical entries.

Desde las contribuciones de Isaac Newton, Gottfried Wilhelm Leibniz, Jacob y Johann Bernoulli en el siglo XVII hasta ahora, las ecuaciones en diferencias y las diferenciales han demostrado su capacidad para modelar satisfactoriamente problemas complejos de gran interés en Ingeniería, Física, Epidemiología, etc. Pero, desde un punto de vista práctico, los parámetros o inputs (condiciones iniciales/frontera, término fuente y/o coeficientes), que aparecen en dichos problemas, son fijados a partir de ciertos datos, los cuales pueden contener un error de medida. Además, pueden existir factores externos que afecten al sistema objeto de estudio, de modo que su complejidad haga que no se conozcan de forma cierta los parámetros de la ecuación que modeliza el problema. Todo ello justifica considerar los parámetros de la ecuación en diferencias o de la ecuación diferencial como variables aleatorias o procesos estocásticos, y no como constantes o funciones deterministas, respectivamente. Bajo esta consideración aparecen las ecuaciones en diferencias y las ecuaciones diferenciales aleatorias. Esta tesis hace un recorrido resolviendo, desde un punto de vista probabilístico, distintos tipos de ecuaciones en diferencias y diferenciales aleatorias, aplicando fundamentalmente el método de Transformación de Variables Aleatorias. Esta técnica es una herramienta útil para la obtención de la función de densidad de probabilidad de un vector aleatorio, que es una transformación de otro vector aleatorio cuya función de densidad de probabilidad es conocida. En definitiva, el objetivo de este trabajo es el cálculo de la primera función de densidad de probabilidad del proceso estocástico solución en diversos problemas basados en ecuaciones en diferencias y diferenciales aleatorias. El interés por determinar la primera función de densidad de probabilidad se justifica porque dicha función determinista caracteriza la información probabilística unidimensional, como media, varianza, asimetría, curtosis, etc., de la solución de la ecuación en diferencias o diferencial correspondiente. También permite determinar la probabilidad de que acontezca un determinado suceso de interés que involucre a la solución. Además, en algunos casos, el estudio teórico realizado se completa mostrando su aplicación a problemas de modelización con datos reales, donde se aborda el problema de la estimación de distribuciones estadísticas paramétricas de los inputs en el contexto de las ecuaciones en diferencias y diferenciales aleatorias. / Ever since the early contributions by Isaac Newton, Gottfried Wilhelm Leibniz, Jacob and Johann Bernoulli in the XVII century until now, difference and differential equations have uninterruptedly demonstrated their capability to model successfully interesting complex problems in Engineering, Physics, Chemistry, Epidemiology, Economics, etc. But, from a practical standpoint, the application of difference or differential equations requires setting their inputs (coefficients, source term, initial and boundary conditions) using sampled data, thus containing uncertainty stemming from measurement errors. In addition, there are some random external factors which can affect to the system under study. Then, it is more advisable to consider input data as random variables or stochastic processes rather than deterministic constants or functions, respectively. Under this consideration random difference and differential equations appear. This thesis makes a trail by solving, from a probabilistic point of view, different types of random difference and differential equations, applying fundamentally the Random Variable Transformation method. This technique is an useful tool to obtain the probability density function of a random vector that results from mapping another random vector whose probability density function is known. Definitely, the goal of this dissertation is the computation of the first probability density function of the solution stochastic process in different problems, which are based on random difference or differential equations. The interest in determining the first probability density function is justified because this deterministic function characterizes the one-dimensional probabilistic information, as mean, variance, asymmetry, kurtosis, etc. of corresponding solution of a random difference or differential equation. It also allows to determine the probability of a certain event of interest that involves the solution. In addition, in some cases, the theoretical study carried out is completed, showing its application to modelling problems with real data, where the problem of parametric statistics distribution estimation is addressed in the context of random difference and differential equations. / Des de les contribucions de Isaac Newton, Gottfried Wilhelm Leibniz, Jacob i Johann Bernoulli al segle XVII fins a l'actualitat, les equacions en diferències i les diferencials han demostrat la seua capacitat per a modelar satisfactòriament problemes complexos de gran interés en Enginyeria, Física, Epidemiologia, etc. Però, des d'un punt de vista pràctic, els paràmetres o inputs (condicions inicials/frontera, terme font i/o coeficients), que apareixen en aquests problemes, són fixats a partir de certes dades, les quals poden contenir errors de mesura. A més, poden existir factors externs que afecten el sistema objecte d'estudi, de manera que, la seua complexitat faça que no es conega de forma certa els inputs de l'equació que modelitza el problema. Tot aço justifica la necessitat de considerar els paràmetres de l'equació en diferències o de la equació diferencial com a variables aleatòries o processos estocàstics, i no com constants o funcions deterministes. Sota aquesta consideració apareixen les equacions en diferències i les equacions diferencials aleatòries. Aquesta tesi fa un recorregut resolent, des d'un punt de vista probabilístic, diferents tipus d'equacions en diferències i diferencials aleatòries, aplicant fonamentalment el mètode de Transformació de Variables Aleatòries. Aquesta tècnica és una eina útil per a l'obtenció de la funció de densitat de probabilitat d'un vector aleatori, que és una transformació d'un altre vector aleatori i la funció de densitat de probabilitat és del qual és coneguda. En definitiva, l'objectiu d'aquesta tesi és el càlcul de la primera funció de densitat de probabilitat del procés estocàstic solució en diversos problemes basats en equacions en diferències i diferencials. L'interés per determinar la primera funció de densitat es justifica perquè aquesta funció determinista caracteritza la informació probabilística unidimensional, com la mitjana, variància, asimetria, curtosis, etc., de la solució de l'equació en diferències o l'equació diferencial aleatòria corresponent. També permet determinar la probabilitat que esdevinga un determinat succés d'interés que involucre la solució. A més, en alguns casos, l'estudi teòric realitzat es completa mostrant la seua aplicació a problemes de modelització amb dades reals, on s'aborda el problema de l'estimació de distribucions estadístiques paramètriques dels inputs en el context de les equacions en diferències i diferencials aleatòries. / Navarro Quiles, A. (2018). COMPUTATIONAL METHODS FOR RANDOM DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/98703 / TESIS

A water distribution system is composed of thousands of pipes of varying materials, sizes, and ages. These pipes experience physical, environmental, and operational factors that cause deterioration and ultimately lead to their failure. Pipe deterioration results in increased break rates, decreased hydraulic capacity, and adverse effects on water quality. Pipe failures result in economic losses to the governing municipality due to loss of service, cost of pipe repair/replacement, damage incurred due to flooding, and disruptions to normal business operations. Inspecting the entire water distribution system for deterioration is difficult and economically unfeasible; therefore, it benefits municipalities to utilize a risk assessment model to identify the most critical components of the system and develop an effective rehabilitation or replacement schedule. This study compared two risk assessment models, a statistically complex model and a simplified model. Based on the physical, environmental, and operational conditions of each pipe, these models estimate the probability of failure, quantify the consequences of a failure, and ultimately determine the risk of failure of a pipe. The models differ in their calculation of the probability of failure. The statistically complex model calculates the probability of failure based on pipe material, diameter, length, internal pressure, land use, and age. The simplified model only accounts for pipe material and age in its calculation of probability of failure. Consequences of a pipe failure include the cost to replace the pipe, service interruption, traffic impact, and customer criticality impact. The risk of failure of a pipe is determined as the combination of the probability of failure and the consequences of a failure. Based on the risk of failure of each pipe within the water distribution system, a ranking system is developed, which identifies the pipes with the most critical risk. Utilization of this ranking system allows municipalities to effectively allocate funds for rehabilitation. This study analyzed the 628-pipe water distribution system in the City of Buellton, California. Four analyses were completed on the system, an original analysis and three sensitivity analyses. The sensitivity analyses displayed the worst-case scenarios for the water distribution system for each assumed variable. The results of the four analyses are provided below. Risk Analysis Simplified Model Complex Model Original Analysis All pipes were low risk All pipes were low risk Sensitivity Analysis: Older Pipe Age Identified 2 medium risk pipes Identified 2 medium risk pipes Sensitivity Analysis: Lower Anticipated Service Life Identified 2 medium risk pipes Identified 9 high risk pipes and 283 medium risk pipes Sensitivity Analysis: Older Pipe Age and Lower Anticipated Service Life Identified 1 high risk pipe and 330 medium risk pipes Identified 111 critical risk pipes, 149 high risk pipes, and 137 medium risk pipes Although the results appeared similar in the original analysis, it was clear that the statistically complex model incorporated additional deterioration factors into its analysis, which increased the probability of failure and ultimately the risk of failure of each pipe. With sufficient data, it is recommended that the complex model be utilized to more accurately account for the factors that cause pipe failures. This study proved that a risk assessment model is effective in identifying critical components and developing a pipe maintenance schedule. Utilization of a risk assessment model will allow municipalities to effectively allocate funds and optimize their water distribution system. Keywords: Water Distribution System/Network, Risk of Failure, Monte Carlo Simulation, Normal Random Variable, Conditional Assessment, Sensitivity Analysis.

[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.