Approximations to the Normal Probability Distribution Function using Operators of Continuous-valued Logic

Abstract

In this study, novel approximation methods to the standard normal probability distribution function are introduced. The techniques presented are founded on applications of certain operators of continuous-valued logic. It is demonstrated here that application of the averaging Dombi conjunction operator to two symmetric Sigmoid fuzzy membership functions results in a function that is identical with Tocher’s approximation to the standard normal probability distribution function. Next, an approximation connected with a unary fuzzy modifier operator is discussed. Namely, the so-called Kappa function is applied for constructing a novel probability distribution function. It is shown here that the asymptotic Kappa function is just the Sigmoid function and the proposed Quasi Logistic probability distribution function can be utilized to approximate the standard normal probability distribution function. It is also explained how the new probability distribution function is connected with the generator function of Dombi operators. The proposed approximation formula is very simple as it has only one constant parameter. It does not include any exponential term, but has a good approximation accuracy and fulfills certain requirements that only a few of the known approximation formulas do.