College of education mathematics department
Application of Bernoulli Inequality
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- Điều hướng trang này:
- 5.4.2. Applications of Bernoulli distribution
- REFERENCES
5.4. Application of Bernoulli Inequality5.4.1. Common application:Exercise 5.19. Let , prove that: Let and . Prove that: Solution: a, We have: By Bernoulli’s inequality, we have: b) The inequality to be proved is equivalent to: By Bernoulli’s inequality, we have: Adding the sides (2) and (3) we get (1), that is, we have things to prove. Exercise 5.20. Let a, b, c be the lengths of the three sides of a triangle. Find the minimum value of: Solution: Lemma: By Bernoulli’s inequality, we have: With is even so Exercise 5.21.Let . Prove that: Solution: Put . Since so . The inequality becomes: By Bernoulli’s inequality, we have: Since so the sign " = " does not occur Hence . Exercise 5.22. Let the positive real numbers a, b, c. Prove that: Solution: Let the given inequality be (*) If one of the three numbers a, b, c is greater than 1, then (*) is true. Indeed, suppose So , On the other hand Hence , but so true. If all three numbers a,b,c are in the interval (0,1), then applying Bernoulli's inequality we have: Similar proof we also have: Adding by (1), (2) and (3) we get 5.4.2. Applications of Bernoulli distributionBernoulli distribution is a form of Binomial distribution where . If then it is converted into Binomial distribution. so the real life application of bernoulli distribution is very rare because N is greater than 1 almost all the study. We consider the applications as following, There are a lot of areas where the application of binomial theorem is inevitable, even in the modern world areas such as computing. In computing areas, binomial theorem has been very useful such a in distribution of IP addresses. With binomial theorem, the automatic distribution of IP addresses is not only possible but also the distribution of virtual IP addresses. Another field that used Binomial Theorem as the important tools is the nation’s economic prediction. Economists used binomial theorem to count probabilities that depend on numerous and very distributed variables to predict the way the economy will behave in the next few years. To be able to come up with realistic predictions, binomial theorem is used in this field. Binomial Theorem has also been a great use in the architecture industry in design of infrastructure. It allows engineers, to calculate the magnitudes of the projects and thus delivering accurate estimates of not only the costs but also time required to construct them. For contractors, it is a very important tool to help ensuring the costing projects is competent enough to deliver profits. REFERENCES[1] Vo Giang Ngai (2002), Inequality topic, Hanoi National University Publishing House. [2] Zdravko Cvetkovski) (1965), Inequalities (Theorems, Techniques and Selected Problems), Informatics Department, European University-Republic of Macedonia. [3] Ovidiu Bagdasar (2006), Inequality and Applications, Faculty of Mathematics and Computer Science, Babes Bolyai University. tải về 344.73 Kb. Chia sẻ với bạn bè của bạn:
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