Approximations for CDF of Standard Normal

Omar Eidous, Areen Bani-Salameh — Tue, 08 Apr 2025 00:00:00 +0200

In this paper, new approximations for the standard normal cumulative distribution function have been proposed. We focused on a set of approximations that take the following form,

0.5(Sqrt(1-exp(-A(z)))

where A(z) is a polynomial function in z consisting of only two terms. The accuracy of the proposed approximations was compared with the accuracy of some existing approximations in the literature that take the same form. The comparison relied on calculating two well-known measures: the maximum absolute error (MAAE) and the mean absolute error (MEAE). The numerical results showed that the proposed approximations outperformed the approximations that were studied.

A Rich Learning Statistical Lesson in the Cauchy

Mohammad Fraiwan Al-Saleh, Arwa Salem Maabreh — Tue, 08 Apr 2025 00:00:00 +0200

Cauchy distribution is a good example of a distribution that has many strange properties and is usually used as a counter example to show that some statistical properties are not always true. In this paper, some hidden properties of the Cauchy distribution are highlighted and studied in details. The focus is to closely look at some very interesting properties of the Cauchy distribution that are rarely mentioned in classrooms. Mentioning these properties in classrooms will have a positive effect on students’ attitude toward learning. One of the main properties of the Cauchy distribution is that it has undefined or infinite moments; the odd moments are undefined while the even moments are infinite. For this distribution, the only way to summarize the data is by ordering them (order Statistics); it is the minimal sufficient statistic. The relation of Cauchy to other distributions such as the Normal, and Uniform are highlighted. The above properties several other properties are discussed in details in this paper. We believe that the content of this manuscript will be a real contribution to educational statistics.

Mutah Journal of Natural, Applied and Health Sciences

Approximations for CDF of Standard Normal

A Rich Learning Statistical Lesson in the Cauchy