Papers by Abdo Abou Jaoude
B P International, 2024
The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov ... more The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the "real" laboratory in R and therefore to compute all the probabilities in the sets R, M, and C. Accordingly, the probability in the whole set C = R + M is constantly equivalent to one independently of the distribution of the input random variable in R, and subsequently the output of the stochastic experiment in R can be determined absolutely in C. This is the consequence of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. Moreover, this pioneering paradigm will be applied to the renowned neutron shielding problem that was used originally in the atomic bomb project and to its stochastic procedures and algorithms in a creative manner.
B P International, 2024
In the current work, we extend and incorporate in the five-axioms probability system of Andrey Ni... more In the current work, we extend and incorporate in the five-axioms probability system of Andrey Nikolaevich Kolmogorov set up in 1933 the imaginary set of numbers and this by adding three supplementary axioms. Consequently, any stochastic experiment can thus be achieved in the extended complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. The purpose here is to evaluate the complex probabilities by considering additional novel imaginary dimensions to the experiment occurring in the “real” laboratory. Therefore, the random phenomenon outcome and result in C = R + M can be predicted absolutely and perfectly no matter what the random distribution of the input variable in R is since the associated probability in the entire set C is constantly and permanently equal to one. Thus, the following consequence indicates that chance and randomness in R is replaced now by absolute and total determinism in C as a result of subtracting from the degree of our knowledge the chaotic factor in the probabilistic experiment. Accordingly, we will apply this pioneering paradigm to the well-known, important, and historical Buffon’s needle technique and to its random algorithms and probabilistic procedures for the computation of in an innovative fashion.
B P International, 2024
The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Ni... more The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set C totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the chaotic factor, the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. We will apply this innovative paradigm to the well-known Monte Carlo techniques and to their random algorithms and procedures in a novel way. Knowing that this groundbreaking paradigm will be illustrated by considering important and famous mathematical problems.
B P International, 2024
Monte Carlo methods were central to the simulations required for the Manhattan Project, though se... more Monte Carlo methods were central to the simulations required for the Manhattan Project, though severely limited by the computational tools at the time. In 1933, Andrey Nikolaevich Kolmogorov established the system of five axioms that define the concept of mathematical probability. This system can be developed to include the set of imaginary numbers and this by adding a supplementary three original axioms. Quantum Monte Carlo, and more specifically Diffusion Monte Carlo methods can also be interpreted as a mean field particle Monte Carlo approximation of Feynman-Kac path integrals. Therefore, any experiment can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. The purpose here is to include additional imaginary dimensions to the experiment taking place in the "real" laboratory in R and hence to evaluate all the probabilities in R, M, and C. Consequently, the probability in the entire set C = R + M is permanently equal to one no matter what the stochastic distribution of the input random variable in R is, therefore the outcome of the probabilistic experiment in C can be determined perfectly. This is due to the fact that the probability in C is calculated after subtracting from the degree of our knowledge the chaotic factor of the random experiment. It is important to state here that one essential and very well-known probability distribution was taken into consideration in the current chapter which is the uniform and discrete probability distribution as well as a specific generator of uniform random numbers, knowing that the original CPP model can be applied to any generator of uniform random numbers that exists in literature. This will yield certainly to analogous results and conclusions and will confirm without any doubt the success of my innovative theory. This novel complex probability paradigm will be applied to the classical probabilistic Monte Carlo numerical methods and to prove as well the convergence of these stochastic procedures in an original way.
B P International, 2024
The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Ni... more The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Hence, we will apply this novel paradigm to Absorbing Markov Chains Theory in order to express it totally and absolutely deterministically in the complex universe C of probabilities.
B P International, 2024
The crucial job of the theory of classical probability is to compute and to assess probabilities.... more The crucial job of the theory of classical probability is to compute and to assess probabilities. A deterministic expression of probability theory will be achieved by the addition of new dimensions to the stochastic experiments. This is the original and novel idea at the foundations of my paradigm. As a matter of fact, since the events outcomes are due to randomness and chance, then the theory of probability is a nondeterministic system in its essence. A deterministic experiment and hence a stochastic event will have a certain result in the complex probability set C after encompassing novel imaginary dimensions to the chaotic experiment occurring in the real set R. Thus, we will be fully knowledgeable to predict the outcome of stochastic experiments that arise in the real world in all stochastic processes if the random event becomes completely predictable. Hence, extending the real probabilities set R to the deterministic complex probabilities set C = R + M by including the contributions of the set M which is the imaginary set of probabilities, is the work that has been accomplished here. Therefore, a novel paradigm of stochastic sciences and prognostic was laid down in which all stochastic phenomena in R were expressed deterministically in C since this extension was found to be successful. I coined this original model by the term: “The Complex Probability Paradigm” or CPP for short. Knowing that it was illustrated and initiated in my previous research publications. Henceforth, this original probability paradigm will be applied in this work to Regular Markov Chains and Processes.
B P International, 2024
In the year 1933, the Russian mathematician Andrey Nikolaevich Kolmogorov put forward the system ... more In the year 1933, the Russian mathematician Andrey Nikolaevich Kolmogorov put forward the system of axioms of modern probability theory. By adding to Kolmogorov’s original five axioms an additional three axioms, this established system can be extended to encompass the imaginary set of numbers. Accordingly, the complex probability set C will be created and which is the sum of its corresponding real probability belonging to the real set R and of its corresponding imaginary probability belonging to the imaginary set M. Thus, all random phenomena do not occur now in the real set R but in the general complex set C that encompasses both R and M. Hence, we take into consideration supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory to evaluate the complex probabilities. This is consequently the objective of this novel paradigm. Subsequently, the outcome of the stochastic experiments that follow any probability distribution in R is now predicted perfectly and totally in C and the corresponding probability in the whole set C is always equal to one. Afterward, it follows that, luck and chance in R is substituted by absolute determinism in C. Therefore, we evaluate the probability of any probabilistic phenomenon in C by subtracting the chaotic factor from the degree of our knowledge of the random system. My groundbreaking Complex Probability Paradigm (or CPP) will be applied to the well-known theory of Markov Chains Transition Matrices in order to express it perfectly and absolutely in a deterministic way in the universe C = R + M as well as to extend it to the probabilities’ universes M and C.
B P International, 2024
In the current manuscript we will apply to the well-known and established theory of Markov Chains... more In the current manuscript we will apply to the well-known and established theory of Markov Chains my groundbreaking and novel Complex Probability Paradigm (or CPP) which will lead to a completely and perfectly deterministic expression of these stochastic processes in the universe of probabilities C = R + M. Consequently, it is important before “probing the depths” of Markov Chains, that we define in this chapter some essential mathematical concepts and fundamental tools and theorems that will be extensively used in the whole manuscript algorithms and C++ programs.
B P International, 2024
We use considerably numerical analysis in science in order to compute the outcome of equations th... more We use considerably numerical analysis in science in order to compute the outcome of equations that cannot be evaluated by exact and ordinary methods like calculus for instance. Therefore, numerical methods were developed to give an approximate outcome of the exact result. Consequently, the differences between the estimated answer and the exact answer will lead to errors. Moreover, the numerical methods adopted determine the magnitude of these errors. Additionally, these numerical methods are dependent on the number of subintervals and iterations taken into account. Furthermore, with respect to deterministic chaos, these discrepancies between the approximate and the exact outcomes in the deterministic methods considered can lead to deterministic chaos although the numerical methods adopted are totally deterministic and completely nonrandom. Hence, we apply in this situation my novel ‘Complex Probability Paradigm (CPP)’ that determines and evaluates the degrees and probabilities of divergence and of convergence in the numerical methods as functions of the number of iterations. Thus, we join here numerical analysis and hence chaos theory to CPP which will be adopted accordingly to prove additionally the convergence of these potential chaotic processes in a novel way by using the law of large numbers. As a central and decisive consequence, we will show that chaos vanishes totally and absolutely in the probability universe C = R + M of my CPP.
B P International, 2024
Numerical analysis is used extensively in science to evaluate the result of equations that cannot... more Numerical analysis is used extensively in science to evaluate the result of equations that cannot be determined by ordinary and exact methods like calculus for example. Hence, numerical methods were designed to give an estimate of the exact outcome. Therefore, this will lead to errors which are the differences between the exact answer and the approximate answer. Additionally, the magnitudes of these errors depend on the numerical method adopted. Moreover, these numerical methods depend on the number of iterations and subintervals considered. Consequently, and according to deterministic chaos theory, these discrepancies between the exact and approximate results in the deterministic methods adopted can lead to deterministic chaos although the numerical methods used are totally nonrandom. Thus, we make use here of my original complex probability paradigm (CPP) that defines the probabilities and degrees of convergence and of divergence in the methods as functions of the iterations number. So, we link CPP with numerical analysis and hence to chaos theory, and the novel model is used thus to prove as well the convergence of these potential chaotic procedures in an original way by using the law of large numbers. As a crucial result, we will prove that chaos vanishes completely and totally in CPP’s probability universe C = R + M.
B P International, 2024
Andrey Nikolaevich Kolmogorov put forward in 1933 the five fundamental axioms of classical probab... more Andrey Nikolaevich Kolmogorov put forward in 1933 the five fundamental axioms of classical probability theory. The original idea in my complex probability paradigm is to add new imaginary dimensions to the experiment real dimensions which will make the work in the complex probability set totally predictable and with a probability permanently equal to one. Therefore, adding to the real set of probabilities R the contributions of the imaginary set of probabilities M will make the event in C = R + M absolutely deterministic. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the novel prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of knowledge, the real and imaginary and complex probabilities, in the probabilities sets R, M, and C. Furthermore, we will apply the new paradigm to numerical analysis and to chaos theory in order to show and to demonstrate that chaos disappears and vanishes completely and absolutely in the complex probability universe C = R + M.
B P International, 2024
The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Ni... more The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set C totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the chaotic factor, the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Accordingly, my purpose here is to link my complex probability paradigm (CPP) to numerical analysis and to chaos theory in order to prove that chaos ceases to exist absolutely in the probability universe C = R + M.
B P International, 2024
Andrey Nikolaevich Kolmogorov put forward in 1933 the five fundamental axioms of classical probab... more Andrey Nikolaevich Kolmogorov put forward in 1933 the five fundamental axioms of classical probability theory. The original idea in my complex probability paradigm is to add new imaginary dimensions to the experiment real dimensions which will make the work in the complex probability set totally predictable and with a probability permanently equal to one. Therefore, adding to the real set of probabilities R the contributions of the imaginary set of probabilities M will make the event in C = R + M absolutely deterministic. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Hence, my purpose is to link my complex probability paradigm to unburied petrochemical pipelines analytic prognostic in the nonlinear damage accumulation case. Consequently, by calculating the parameters of the novel prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of knowledge, the complex probability, the system failure and survival probabilities, and the remaining useful lifetime probability, after that a pressure time t has been applied to the pipeline and which are all functions of the system degradation subject to random effects. Furthermore, we will apply the new paradigm to my novel ‘Dynamic Logic’ model.
B P International, 2024
The set of imaginary numbers is taken into account by extending the probability system of five ax... more The set of imaginary numbers is taken into account by extending the probability system of five axioms of Andrey Nikolaevich Kolmogorov which was put forward in 1933. This is achieved by adding three new and supplementary axioms. Hence, any random experiment can thus be performed in the extended complex probability set C = R + M which is the sum of the real set R of real probabilities and the imaginary set M of imaginary probabilities. The objective here is to determine the complex probabilities by encompassing and considering additional new imaginary dimensions to the event that occurs in the “real” laboratory. The outcome of the stochastic phenomenon in C can be foretold perfectly whatever the probability distribution of the input random variable in R is since the corresponding probability in the whole set C is permanently and constantly equal to one. Thus, the consequence that follows indicates that randomness and chance in R is substituted now by absolute determinism in C. This is the result of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. This novel complex probability paradigm will be applied to a newly defined logic that I called “Dynamic Logic”.
B P International, 2024
The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Ni... more The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set C totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the chaotic factor, the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Accordingly, my purpose here is to link my complex probability paradigm The Paradigm of Complex Probability, Prognostic, and Dynamic Logic The Paradigm of Complex Probability and the Novel Dynamic Logic-The Model 2 to logic. Hence, after adding the time dimension, we will apply this novel paradigm to a newly defined logic that I called 'Dynamic Logic'.
B P International, 2024
I wish that the Theory of Metarelativity will be a fascinating mathematical model and work of sci... more I wish that the Theory of Metarelativity will be a fascinating mathematical model and work of science that attempts to explain, among other things, the existence and the nature of dark matter, of dark energy, of tachyons which are faster-than-light particles etc. It is crucial to state here that Metarelativity does not destroy Albert Einstein’s special and general theories of relativity at all but on the contrary, it proves their veracity and truth and tries to be a successful extension of Einstein’s theories to the realm of higher-dimensional and imaginary and complex matter, energy, time, length, entropy etc. starting from the well-known relativistic Lorentz transformation. We have to wait certainly that it will be proved experimentally to decide its ultimate fate.
B P International, 2024
Calculating probabilities is the crucial task of classical probability theory. Adding supplementa... more Calculating probabilities is the crucial task of classical probability theory. Adding supplementary dimensions to nondeterministic experiments will yield a deterministic expression of the theory of probability. This is the novel and original idea at the foundations of my complex probability paradigm. As a matter of fact, probability theory is a stochastic system of axioms in its essence; that means that the phenomena outputs are due to randomness and chance. By adding novel imaginary dimensions to the nondeterministic phenomenon happening in the set R will lead to a deterministic phenomenon and thus a stochastic experiment will have a certain output in the complex probability set and total universe G = C. If the chaotic experiment becomes completely predictable then we will be fully capable to predict the output of random events that arise in the real world in all stochastic processes. Accordingly, the task that has been achieved here was to extend the random real probabilities set R to the deterministic complex probabilities set and total universe G = C = R + M and this by incorporating the contributions of the set M which is the complementary imaginary set of probabilities to the set R. Consequently, since this extension reveals to be successful, then an innovative paradigm of stochastic sciences and prognostic was put forward in which all nondeterministic phenomena in R was expressed deterministically in C. Furthermore, this model was linked in Chapter Five to my Theory of Metarelativity which takes into consideration faster-than-light matter and energy. This is what I named ‘The Metarelativistic Complex Probability Paradigm (MCPP)’ where some of its essential and crucial consequences will be presented in the present chapter.
B P International, 2024
In this work and in the following sections of this chapter we will consider three simplified mode... more In this work and in the following sections of this chapter we will consider three simplified models of MCPP then present at the end the final and most general model. Each model is an enhanced and a wider model than the previous one. In all the four models, we will consider the velocities of the bodies moving in G to be random variables that follow certain probability distributions (PDFs) and certain corresponding cumulative probability distribution functions (CDFs) in both the subluminal universe G1 and the superluminal metauniverse G2. I have followed this methodology in order to develop gradually and systematically the MCPP paradigm and in order to reach the final and most general model of MCPP that can be adopted in any possible and imaginable situation. Even the deterministic case which is a special case of the general random CPP was also presented and considered in order to show that MCPP is valid everywhere either in the deterministic or in the random case. Consequently, and in each model, we evaluate the associated real, imaginary, and complex probabilities as well as all the related MCPP parameters in the probabilities sets R, M, and C hence in the universes G1, G2, G3, and G. Thus, we connect successfully CPP with Metarelativity to unify both theories in a general and a unified paradigm that we called MCPP.
B P International, 2024
The Theory of Metarelativity is a system of equations written to take into consideration addition... more The Theory of Metarelativity is a system of equations written to take into consideration additional effects in the universe and about the matter inside it. The study begins with Albert Einstein’s theory of special relativity and it develops a system of equations which lead us to further explanations and to a new physics paradigm. Like special relativity which was created in 1905 and then expanded later to general relativity to explain, among other things, the aberration in the motion of the planet Mercury and the gravitational lenses, metarelativity explains many phenomena like for example the nature of dark matter laying inside and outside galaxies and in the universe and the existence of superluminal particles or tachyons and their corresponding dark matter and dark energy. Metarelativity is a work of pure science which encompasses mathematics and fundamental physics. All the explanations are deduced from a new system of equations called the metarelativistic transformation that will be proven mathematically and explained physically. The facts and experiments that are noted in this theory come from a large series of astronomical observations taken far later than 1905 till now by reliable observatories in the world.
B P International, 2024
Computing probabilities is the main work of classical probability theory. Adding new dimensions t... more Computing probabilities is the main work of classical probability theory. Adding new dimensions to the stochastic experiments will lead to a deterministic expression of probability theory. This is the original idea at the foundations of this work. Actually, the theory of probability is a nondeterministic system in its essence; that means that the events outcomes are due to chance and randomness. The addition of novel imaginary dimensions to the chaotic experiment occurring in the set R will yield a deterministic experiment and hence a stochastic event will have a certain result in the complex probability set C. If the random event becomes completely predictable then we will be fully knowledgeable to predict the outcome of stochastic experiments that arise in the real world in all stochastic processes. Consequently, the work that has been accomplished here was to extend the real probabilities set R to the deterministic complex probabilities set C = R + M by including the contributions of the set M which is the imaginary set of probabilities. Therefore, since this extension was found to be successful, then a novel paradigm of stochastic sciences and prognostic and physics was laid down in which all stochastic phenomena in R was expressed deterministically.
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Papers by Abdo Abou Jaoude
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods
are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.
Furthermore, in physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures. Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, the evaluation of multidimensional definite integrals with complicated boundary conditions. In application to systems engineering
problems (space, oil exploration, aircraft design, etc.), Monte Carlo–based
predictions of failure, cost overruns, and schedule overruns are routinely better than human intuition or alternative “soft” methods. In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation.
This volume illustrates the use of Monte Carlo Methods when applied to solve specific problems. It delves deeply into this exciting, profound, and modern field of mathematics and knowledge. The book discusses some fundamental aspects of applied Monte Carlo Techniques and, as such, it is a useful resource for scholars, researchers, and undergraduate and graduate students in pure and applied mathematics, physical sciences, engineering and technology, computer science, numerical analysis, scientific computing, and science in general.
statistical phenomena in the following areas: sampling theory, estimation
theory, tests of hypotheses and significance, curve-fitting, regression and
correlation, analysis of variance, nonparametric tests, and analysis of time
series, in addition to the presentation of my “Complex Probability Paradigm” applied to Monté Carlo Methods and to the Central Limit Theorem. As such, the book will be of interest to all scholars, researchers, and undergraduate and graduate students in mathematics, statistics, computer science, and science in general.
statistical phenomena in the following areas: sampling theory, estimation
theory, tests of hypotheses and significance, curve-fitting, regression and
correlation, analysis of variance, nonparametric tests, and analysis of time
series, in addition to the presentation of my “Complex Probability Paradigm” applied to Monté Carlo Methods and to the Central Limit Theorem. As such, the book will be of interest to all scholars, researchers, and undergraduate and graduate students in mathematics, statistics, computer science, and science in general.
Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the “real” laboratory in R and therefore to compute all the probabilities in the sets R, M, and C. Accordingly, the probability in the whole set C = R + M is constantly equivalent to one independently of the distribution of the input random variable in R, and subsequently the output of the stochastic experiment in R can be determined absolutely in C. This is the consequence of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. We will apply this innovative paradigm to the well-known Central Limit Theorem and to prove as well its convergence in a novel way.
probability distribution of the input random variable in R is since the corresponding probability in the whole set C is permanently and constantly equal to one. Therefore, the consequence that follows indicates that randomness and chance in R is substituted now by absolute determinism in C. This novel complex probability paradigm will be implemented to the field of prognostic based on reliability, hence to the concepts of the system remaining useful lifetime (RUL) and degradation. Additionally, the First-Order Reliability Method (FORM) analysis will be applied to Young’s modulus to illustrate my original and innovative paradigm.