Bernoullis inequality and the number \e\ by joseph wiener pan american university this article originally appeared in. Please subscribe here, thank you proof of bernoulli s inequality using mathematical induction. Finally, invent a random variable and a distribution such that, prx 10ex 1 10. Hoeffdings inequality is a generalization of the chernoff bound, which applies only to bernoulli random variables, and a special case of the azumahoeffding inequality and the mcdiarmids inequality. A simple proof of bernoull is inequality sanjeev saxena bernoullis inequality states that for r 1 and x 1. The amgm inequality is equivalent to the bernoulli inequality. Many of them have been proved by famous mathematicians and named after them.
One can find in the literature several elementary different proofs of inequality 1. Using induction to prove bernoullis inequality youtube. Much earlier, isaac barrow 1670 and jacob bernoulli. Let x be a random variable with distribution f, and let u be an independent uniform0,1 random variable. A simple proof of bernou llis inequality sanjeev saxena dept. Bernoullip then and x n n 1 p n i1 x i then, varx n varx 1n p1 pnand pjx n pj varx n 2 p1 p n 1 4n 2 since p1 p 1 4 for all p. Convert inequality statements to equations and workwiththeequations. If xand yare real numbers of the same sign xand yare either both positive or both negative, andx 0. Bernoullis inequality and the number \e\ mathematical. Around bernoullis, 19 th all polish conference on history of mathematics june 610, 2005, zamosc, lublin technical university publisher, lublin, 2006, 3162 in polish. If n 1 the result is clear since any cupcake has the same avor as. A pdf copy of the article can be viewed by clicking below.
In this latter case the proof easily follows from chebychevs inequality. X, and so by hypothesis the unconditional mean is ee1u xjx. This document is aimed to provide clear and complete proof for some inequalities. By using methods on the theory of majorization, some generalizations of bernoulli s inequality were established by shi 16. What is wrong with the following proof that all cupcakes have the same avor.
The amgm inequality is equivalent to the bernoulli. Some strengthened forms of bernoulli s inequality are established. In this paper, a new proof of bernoullis inequality via the dense concept is given. In particular, for any probability distribution f on. The proof is only based on the fact that for any n nonnegative numbers, geometric mean can not exceed. Moreover, some equivalent relations between this inequality and other. Calculate the inequality of any given function through bernoullis inequality mathematical induction calculator when x value and power r is known and apply the formula. The strict inequality is not discussed in the following. Bernoulli s inequality is one of the most distinguished inequalities. Since the copy is a faithful reproduction of the actual journal pages, the article may not begin at the top of the first page. It is similar to, but incomparable with, the bernstein inequality, proved by sergei bernstein in 1923. Maligranda, bernoulli inequality over 300 years of history, in.
In particular, we obtain harmonicgeometricarithmetic mean inequality and. A gentle introduction to concentration inequalities. A new look at bernoullis inequality american mathematical society. Please subscribe here, thank you proof of bernoullis inequality using mathematical induction. If the exponent r is even, then the inequality is valid for all real numbers x. A simple proof of bernoullis inequality sanjeev saxena dept. Suppose a, b, c, and dare positive real numbers, ab, and cd.
Inequality and the mean inequalities107 11 newtons inequality. Although bernoulli deduced that pressure decreases when the flow. Pdf a generalization of bernoullis inequality researchgate. Maclaurins inequality and a generalized bernoulli inequality. The bernoulli boys bernoullis inequality is named after jacques bernoulli, a swiss mathematician who used it in a paper on in. Some strengthened forms of bernoullis inequality are established. Sanjeev saxena bernoullis inequality states that for r. Concentration inequalities are inequalities that bound prob. Cherno bounds, and some applications 1 preliminaries. Today, bernoullis law of large numbers 1 is also known as the weak law of large numbers. Some classical inequalities among all inequalities, there is a number of wellknown classical inequalities.
A generalization of bernoullis inequality fiu digital commons. Chord lengths, discriminants of cyclotomic fields and reducibility of cyclotomic polynomials modulo primes b. For, the following generalizations of bernoulli inequality are valid for real exponents. In the next theorems, by applying methods on the utility theory, some. Proof of bernoullis inequality mathematics stack exchange. My favorite way of proving bernoulli is to use jensen inequality. One benefit of finding an interpolated bernoullis inequality is that it will lead to a new proof of maclaurins inequality. Moreover, some equivalent relations between this inequality and other known inequalities are tentatively linked. Here are some special cases of the power mean inequality. Pdf some equivalent forms of bernoullis inequality. Bernoullis inequality is one of the most distinguished inequalities. In this paper, a new proof of bernoulli s inequality via the dense concept is given. The proof applies to infiniterange models on arbitrary locally finite transitive infinite graphs.
We provide a new proof of the sharpness of the phase transition for bernoulli percolation and the ising model. A basic understanding of probability theory and set algebra might be required of the reader. Michel goemans 1 preliminaries before we venture into cherno bound, let us recall chebyshevs inequality which gives a simple bound on the probability that a random variable deviates from its expected value by a certain amount. Pdf the amgm inequality is equivalent to the bernoulli. This importantly shows that markovs inequality is tight, because we could replace 10 with tand use bernoulli1, 1t, at least with t 1. Chebyshevs inequality can be derived as a special case of markovs inequality. The bernoulli distribution is an example of a discrete probability distribution. In this paper, some generalizations of the bernoullis inequality are derived by using methods from the utility theory. Finally applying hoe dings inequality gives the following bound. Proof of bernoullis inequality using mathematical induction. Next, we will prove bernoullis inequality by means of. Since the arithmeticgeometric mean inequality is interpolated. The amgm inequality is equivalent to the bernoulli inequality article pdf available in the mathematical intelligencer 341 march 2012 with 1,033 reads how we measure reads.
In fluid dynamics, bernoullis principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluids potential energy. Pdf we prove a generalization of bernoullis inequality and we apply this generalization to sharpen certain weierstrass product inequalities find, read and. We prove the following generalization of bernoullis inequality. The liquid enters the pipe with a normal velocity v 1 and at a height h 1 above the reference level earths surface.