The paradigm of complex probability and Chebyshev’s inequality

Abstract

The 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 C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. 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 C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that, chance and luck in R is replaced by total determinism in C. 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 C. 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 C.