Yakışıklı erkek tatil için bir beldeye gidiyor burada kendisine türk Porno güzel bir seksi kadın ayarlıyor Onunla beraber otel odasına gidiyorlar Otel odasına rokettube giren kadın ilk önce erkekle sohbet ederek işi yavaş halletmeye çalışıyor sex hikayeleri Kocası fabrikatör olan sarışın Rus hatun şehirden biraz uzak olan bir türk porno kasabaya son derece lüks bir villa yaptırıp yerleşiyor Kocasını işe gönderip mobil porno istediği erkeği eve atan Rus hatun son olarak fotoğraf çekimi yapmak üzere türk porno evine gelen genç adamı bahçede azdırıyor Güzel hatun zengin bir iş adamının porno indir dostu olmayı kabul ediyor Adamın kendisine aldığı yazlık evde sikiş kalmaya başlayan hatun bir süre sonra kendi erkek arkadaşlarını bir bir çağırarak onlarla porno izle yapıyor Son olarak çağırdığı arkadaşını kapıda üzerinde beyaz gömleğin açık sikiş düğmelerinden fışkıran dik memeleri ile karşılayıp içeri girer girmez sikiş dudaklarına yapışarak sevişiyor Evin her köşesine yayılan inleme seslerinin eşliğinde yorgun düşerek orgazm oluyor

Cauchy-Schwarz Inequality: Another Proof

Kunroo MH1*, Aggarwal DR2

1Sir Ratan Tata Trust Fellowship Institute of Economic Growth, Delhi University Enclave, New Delhi, India

2School of Management, The NorthCap University, Sector 23A, Gurgaon 122017, India

*Corresponding Author:
Kunroo MH
Young Social Scientist under the scheme of Sir Ratan Tata Trust Fellowship B-9 (Second Floor), Institute of Economic Growth, University of Delhi (North Campus) Enclave, Delhi, India, 110007
Tel:
911127667288.
E-mail:
mhkunroo@gmail.com

Received date: 15/04/2017; Accepted date: 29/04/2017; Published date: 30/04/2017

Visit for more related articles at Research & Reviews: Journal of Statistics and Mathematical Sciences

Abstract

The Cauchy-Schwarz inequality is one of the most famous inequalities in mathematics. Over the period of time, it has been manifested in various forms and there are many different ways to prove this inequality. In this paper, we achieve one more proof to this inequality by making use of two basic statistical techniques, namely the coefficient of correlation and the coefficient of determination.

Keywords

Cauchy-Schwarz inequality, another proof, Probability, Calculus

Introduction

Over the years, the Cauchy-Schwarz inequality has made a prominent place in applied sciences. This inequality was first time jotted down in the form of finite sums by French mathematician Augustin-Louis Cauchy (1789-1857), in his work Cours d’analyse de l’École Royal Polytechnique, first published in 1821. Later on, in 1859, Cauchy’s former student from Russia, Viktor Yakovlevich Bunyakovsky (1804-1889), proved the inequality for infinite sums, and was written first time as integrals. The proof was published in the journal Mémoires de l’Académie Impériale des Sciences de St-Pétersbourg. Again, in 1888, Karl Hermann Amandus Schwarz (1843-1921), unaware of the work of Bunyakovsky, presented an independent proof of Cauchy’s inequality in integral form. Such an evolution of the inequality is the main reason behind its several names in literature, for example Cauchy- Schwarz, Schwarz, and Cauchy-Bunyakovsky-Schwarz inequality. The inequality, along with its extensions and generalizations, has been applied in diverse fields of mathematics and physics, such as partial differential equations, multivariable calculus, linear algebra, geometry, probability theory, operator theory and matrix analysis, uncertainty principles of Heisenberg and Schrӧdinger, etc [1-5]. This wide variety of applications of Cauchy-Schwarz inequality created its numerous manifestations and forms. The most prominent ones are:

The classical Cauchy-Schwartz Inequality states that if ai, bi (i=1, 2, 3,…..n) are real quantities i.e., ai, bi equation then,

equation

The various forms of Cauchy-Schwarz inequality has resulted in a number of proofs. In fact, there are many different ways to prove this inequality [6,7]. For instance, Wu and Wu [8] present twelve different proofs for the classical Cauchy-Schwarz inequality. In this paper, we make use of statistical techniques, namely the coefficient of correlation (r) and coefficient of determination (r2) to achieve this proof. To the best of our knowledge, such a kind of proof of this inequality has not been provided till now. Accordingly, it serves a good contribution to the existing literature.

Proof

Suppose X and Y are two random variables and, assuming their variances are positive, then the

correlation of (X, Y) is defined as

equation

Squaring the coefficient of correlation gives coefficient of determination, which means

equation

equation

Hence proved.

Acknowledgements

The first author is mainly responsible for coming up with such an idea/proof. The second author executed the idea and put it in the present shape and form. The usual disclaimers apply.

References