There are many beautiful formulas for π (see for example [4]). The purpose of this note is to introduce an alternate derivation of Wallis’s product formula, equation (1), which could be covered in a first course on probability, statistics, or number theory. We quickly review other famous formulas for π, recall some needed facts from probability, and then derive Wallis’s formula. We conclude by ...

by repeated partial integration. The topic is usually reserved for more advanced calculus courses. The purpose of this note is to show that (1) can be derived using only the mathematics taught in elementary school, that is, basic algebra, the Pythagorean theorem, and the formula π · r 2 for the area of a circle of radius r . Viggo Brun gives an account of Wallis’s method in [1] (in Norwegian). ...

The m-clover is the plane curve defined by the polar equation rm/2 = cos (m2 θ). In this article we extend a well-known derivation of the Wallis product to derive a generalized Wallis product for arc lengths of m-clovers.

This equation becomes Wallis’ product when p = 0 and Vieta’s formula as p → ∞. It is surprising that such a connection between the two products was not discovered earlier. The collection [1] contains both original papers of Vieta and Wallis as well as other fundamental papers in the history of π. Indeed, there are many good historical sources on π. The text by P. Eymard and J. P. Lafon [6] is a...

The effects of the violations normality and homogeneity variances assumptions on power one-way ANOVA F-test is studied in this paper. Simulation experiments were conducted to compare parametric with non-parametric Kruskal‒Wallis (KW) test normal/non-normal, equal/unequal scenarios sample group means. Each these 184 simulation was replicated N = 1000 times obtained for both F KW tests. Shapiro‒W...

Specific statistical tests are used when the null hypothesis (H0) is to be tested using nonparametric nominal or ordinal data. With nominal data, experimental results are expressed by proportions or frequencies. Chi-square or related tests (the Fisher's exact test or the rows by columns test) are appropriate for testing H0 with nominal data. Ordinal data permit arrangement of statistical result...

Coal beneficiation processes are often based on gravitational enrichment techniques. Therefore, various types of jigs used in this area. A number factors determine the quality final product. proper and systematic monitoring such a process is necessary. The present work examined effect hutch water amount processed coal (system capacity) yield ash grade individual density–size fractions. statisti...