A Simple Approximation for Normal Distribution Function

Edous, Medhat; Eidous, Omar

doi:10.13189/ms.2018.060401

Cited by 5 publications

(2 citation statements)

References 2 publications

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“…where p ∈ [0, 1] is an imbedding parameter. When we put p → 1, then equation (3) corresponds to (2), and (5) becomes the approximate solution of (7), that is…”

Section: Approximation By Hpmmentioning

confidence: 99%

“…Unfortunately, Φ(x) cannot be expressed in a closed form in terms of elementary functions for all values of x which would be useful for the practical needs, many numerical approximations for Φ(x) are known. A number of approximate functions for CDF have been proposed (see, for example, [2,3], and references therein). Let us represent CDF (3) in the following form…”

Section: Introductionmentioning

confidence: 99%

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On HPM Approximations for the Cumulative Normal Distribution Function

Shchigolev¹

2019

ujcmj

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In this paper, some new approximations to the cumulative distribution function of the standard normal distribution via the He's homotopy perturbation method are proposed. There are several methods which provide an approximation of the integral in the formula for the cumulative distribution function by different numerical methods. For the same purpose, we first establish a differential equation of the second order that the cumulative distribution function satisfied subjected with the certain initial conditions. Then we apply the Homotopy Perturbation Method to solve the Cauchy problem for the governing equation. As well known, the result of solving an equation by this method and the convergence rate greatly depend on the choice of homotopy applied. Therefore, we consider two cases in this work. In one case, we construct the homotopy from the idea of simplicity. In the next case, we just follow the procedure of the general approach proposed early. As a result, we obtain several approximations which can be are easily calculated and are better than some other approximations. Numerical comparison shows that our approximations are very accurate.

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“…where p ∈ [0, 1] is an imbedding parameter. When we put p → 1, then equation (3) corresponds to (2), and (5) becomes the approximate solution of (7), that is…”

Section: Approximation By Hpmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On HPM Approximations for the Cumulative Normal Distribution Function

Shchigolev¹

2019

ujcmj

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Wahrscheinlichkeitsrechnung

Proske

2023

Baustatistik for Beginners

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New Mathematical Approximations for the Cumulative Normal Distribution Function

Etesami,

Madadi,

Keynia

et al. 2024

Preprint

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The normal cumulative distribution function and its derivatives, such as the error function, the Q-function, and the Mills ratio, are widely used in engineering, mathematics, statistics, computer science, diffusion theory, communication theory, physics, and chemistry. However, their non-closed form nature has led to the development of new approximations with varying levels of accuracy and complexity. These new approximations are often more accurate; nevertheless, they can also be more complex, which may limit their practical utility. In this article, a new approach for approximating is proposed. which combines Taylor series expansion and logistic function to create an initial approximation, to enhance the accuracy of the initial approximation, the Hunter-Prey Optimization algorithm is utilized to minimize both the maximum absolute error and the mean absolute error, leading to a significantly more precise approximation. Furthermore, this algorithm is employed to enhance the accuracy of other existing approximations introduced by researchers. The results showed that the improved approximations have much higher accuracy. To show the effectiveness of the new findings of this article, two case studies with applications are presented.

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A Simple Approximation for Normal Distribution Function

Cited by 5 publications

References 2 publications

On HPM Approximations for the Cumulative Normal Distribution Function

On HPM Approximations for the Cumulative Normal Distribution Function

Wahrscheinlichkeitsrechnung

New Mathematical Approximations for the Cumulative Normal Distribution Function

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