Abstract
ABSTRACTThe system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set , which is the sum of the real set with its corresponding real probability, and the imaginary set with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set instead of the real set . The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set is always equal to one and the outcome of the random experiments that follow any probability distribution in is now predicted totally in . Subsequently, it follows that, chance and luck in is replaced by total determinism in . Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in . This novel complex probability paradigm will be applied to the established theorem of Pafnuty Chebyshev's inequality and to extend the concepts of expectation and variance to the complex probability set .
Highlights
One of the branches of mathematics is the probability theory, which is devoted to probability and to the analysis of random phenomena
In order to have a certain prediction of any random event, it is necessary to work in the complex set C in which the chaotic factor is quantified and subtracted from the computed degree of knowledge to lead to a probability in C equal to one (Pc2 = degree of our knowledge (DOK) − Chf = DOK + magnitude of the chaotic factor (MChf) = 1)
The magnitude of the chaotic factor MChf We have from the complex probability paradigm (CPP) MChf = −2iPrPm = 2PrPm/i = 2Pr(1 − Pr) = 2Pr − 2Pr2 where i2 = −1 and i = − 1 . i
Summary
The famous mathematician Bernoulli, half a century later, showed a sophisticated grasp of probability He showed, facility with permutations and combinations and discussed the concept of probability with examples beyond the classical definition, such as personal, judicial, and financial decisions, and showed that probabilities could be estimated by repeated trials with uncertainty diminished as the number of trials increased (Wikipedia, Probability Measure, Probability Axioms, Chebyshev’s Inequality). [Marquis Pierre-Simon de Laplace, A Philosophical Essay on Probabilities (Wikipedia, Probability Theory)] This classical definition provides what would be the ultimate description of probability. The more advanced ‘measure theory’, which is based on the treatment of probability, covers both the discrete, the continuous, any mix of these two and more Since they well describe many natural or physical processes, certain random variables occur very often in probability theory. I conclude the work by doing a comprehensive summary in Section 9, and present the list of references cited in the current research work
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