Generalized probability density function and applications to the experimental data in electrical circuits and systems

Abstract

Summary The mathematical sciences, and particularly probabilistic and statistical methods, are key to understanding the dependencies of the systems. The purpose of this paper is to encourage a wider recognition by engineers of a new generalized principle which in its mathematical form is a powerful instrument for the solution of practical problems. Generalized probability density function was introduced to permit analysis without pre‐knowledge of the source of the data. The fundamental principles are extended to apply the most related engineering applications without the need to know the type of source generating the data. The generalized model presented eliminates preliminary work in engineering problems. The proposed model introduces an exponential density function to produce a direct solution to randomly varying data. The exponential density function is fully compatible with applications containing randomly distributed data. The success of the generalized model presented is due to the calculated parameters in the exponential density function. The method is applied to various problems chosen from the field of engineering with great success.

Further theories on application of new generalized probability density function and its applications

“…After very encouraging results of this PDF function (1) in the paper, 1 many further statistical and probabilistic theories should be studied for improving and enlarging the applicability of the function (1). So in this paper, the nth moment of PDF ( 1) is derived in Section 2.…”

“…Normally, it is very difficult to fit the data set accurately without the preknowledge of the behavior of the experimental results. In the paper, 1 the new generalized probability density function has been introduced as 𝑓(𝑥; 𝛼, 𝛽, 𝑞) = 1 Γ(𝑞 + 1) (2𝛼𝑥 + 𝛽) ( 𝛼𝑥 2 + 𝛽𝑥 ) 𝑞 𝑒 −(𝛼𝑥 2 +𝛽𝑥) , 𝑥 > 0, 𝛼, 𝛽 ≥ 0, Re(𝑞) > −1.…”

“…Consequently, it was shown that the probability density function (1) reduces some well-known distributions such as half normal, exponential, Rayleigh, Weibull, and gamma. Additionally, the PDF function (1) can present different probability functions after deriving the necessary values of 𝛼, 𝛽 ≥ 0 and Re(𝑞) > −1. In the paper, 1 the newly introduced PDF function (1) is successfully applied in electric and electronic engineering.…”

“…Additionally, the PDF function (1) can present different probability functions after deriving the necessary values of 𝛼, 𝛽 ≥ 0 and Re(𝑞) > −1. In the paper, 1 the newly introduced PDF function (1) is successfully applied in electric and electronic engineering. In the paper, 1 the unknown parameters 𝛼, 𝛽 ≥ 0 in Equation (1) in the applications are found by using the probabilistic expectation.…”

“…In the paper, 1 the newly introduced PDF function (1) is successfully applied in electric and electronic engineering. In the paper, 1 the unknown parameters 𝛼, 𝛽 ≥ 0 in Equation (1) in the applications are found by using the probabilistic expectation. Additionally, another important parameter 𝑞 is estimated empirically by using the Kolmogorov-Smirnov (K-S) theorem.…”