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have �0.5 ≤ Chf < 0, 0 < MChf ≤ 0.5, and 0.5 ≤ DOK < 1. Notice that during this whole process, we have always Pc<sup>2</sup> = DOK – Chf = DOK + MChf =1= Pc, which means that the phenomenon which looked to be stochastic and random in the set R is now certain and deterministic in the set C ¼ R þM, and this after the addition of the contributions of M to the phenomenon occurring in R and thus after subtracting and eliminating the chaotic factor from the degree of our knowl-

corresponding to each instant t have been evaluated, in addition to the probability of RUL after a pressure cycles time t, which are all functions of the stochastic degradation jump. Consequently, at each instance of t, all the novel CPP parameters D, RUL, Pr, Pm, Pm=i, DOK, Chf, MChf, Pc, and Z are certainly and perfectly predicted in the complex probability set C with Pc maintained as equal to 1 constantly and permanently. Furthermore, using all these illustrated simulations and drawn graphs all over the whole research work, we can quantify and visualize both the certain knowledge (expressed by DOK and Pc) and the system chaos and random effects (expressed by Chf and MChf) of the pipeline system. This is definitely very fascinating, fruitful, and wonderful and proves once again the advantages of extending the five probability axioms of Kolmogorov and thus the novelty and benefits of this original field in prognostic and applied mathematics that can be

As a prospective and future work and challenges, and concerning some applications to practical engineering, it is planned to more elaborate the original created prognostic paradigm and to implement it to a varied set of nondeterministic and dynamic systems like vehicle suspension systems and offshore and buried petrochemical pipes which are under the influence of fatigue and in the cases of

nonlinear and linear damage accumulation. Furthermore, we will apply also CPP to other random experiments in classical probability theory and in stochastic processes and to the field of prognostic in engineering using the first order reliability method (FORM) as well as to the random walk problems which have enormous applications

<sup>2</sup> ¼ �1 and <sup>i</sup> <sup>¼</sup> ffiffiffiffiffiffi

�<sup>1</sup> <sup>p</sup>

in physics, in economics, in chemistry, in applied and pure mathematics.

No potential conflict of interest was reported by the author.

Pr system failure probability, probability in the real set R

Pm system survival probability in M, probability in the imaginary set M corresponding to the real probability in R

Pc probability in the complex set C, probability of an event in R with

R The set of real probabilities of events M The set of imaginary probabilities of events C The set of complex probabilities of events

i The imaginary number where i

Pm/i system survival probability in R

its associated event in M

EKA extended Kolmogorov axioms CPP Complex probability paradigm

Prob any event probability

edge. Moreover, the probabilities of the system survival and of failure

called verily "The Complex Probability Paradigm."

Fault Detection, Diagnosis and Prognosis

Conflict of interest

Nomenclature

98

Abdo Abou Jaoude

Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaize, Lebanon

\*Address all correspondence to: abdoaj@idm.net.lb

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
