Jensen inequality exercises
WebMar 24, 2024 · Jensen's Inequality. If , ..., are positive numbers which sum to 1 and is a … WebOne of the simplest examples of Jensen's inequality is the quadratic mean - arithmetic …
Jensen inequality exercises
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WebOct 6, 2024 · This page titled 6.E: Solving Equations and Inequalities (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. WebJensen's Inequality is an inequality discovered by Danish mathematician Johan Jensen in 1906. Contents 1 Inequality 2 Proof 3 Example 4 Problems 4.1 Introductory 4.1.1 Problem 1 4.1.2 Problem 2 4.2 Intermediate 4.3 Olympiad Inequality Let be a convex function of one real variable. Let and let satisfy . Then If is a concave function, we have: Proof
WebExercise 1.2 1. (Conditional Jensen inequality) Using the property that a convex func-tion … WebArithmetic and geometric means satisfy a famous inequality, namely that the geometric …
Webity known as Jensen’sinequality (we will see this very important inequality later in our derivation of the EM algorithm). Jensen’s inequality states the following: if f : R→ Ris a convex function, meaning that f is bowl-shaped, then f(E[Z]) ≤ E[f(Z)]. The simplest way to remember this inequality is to think of f(t) = t2, and WebExercise 1.2 1. (Conditional Jensen inequality) Using the property that a convex func-tion ψ: R → R admits the representation ψ(x) = sup l∈L ψ l(x), where L ψ is the set of all linear functions l≤ ψ, show that ψ(E[X G]) ≤ E[ψ(X) G], (Please note that some integrability conditions are required: it is left to you to figure them out)
WebSep 1, 2024 · 3 In his probability book Bauer proves the following version of Jensen's inequality: Proposition. Let X be an integrable random variable taking values in an open interval I ⊂ R, and let q be a convex function on I. If q ∘ X is integrable, then q ( E ( X)) ≤ E ( q ∘ X). Now am asked to prove that the result holds for an arbitrary interval, e.g.
Webget good estimates for the mean and variance. We can use these concentration inequalities to bound the probability that we wait too long for a bus knowing just those two quantities and nothing else! 6.1.1 Markov’s Inequality We’ll start with our weakest inequality, Markov’s inequality. This one only requires us to know the mean, and ... tammy\u0027s notary service boswell paWebJensen’s inequality below). Method 4: (Lagrange multipliers) Consider the function of n vari- ... This is equivalent to the Cauchy-Schwarz inequality. As an exercise, consider the case n = 2 and find a relation between the Cauchy-Schwarz and the AM-GM inequality. 0.5. Various Putnam Exam problems involving inequalities: tammy\u0027s nails redding caWebget good estimates for the mean and variance. We can use these concentration … tybee island police reportshttp://cs229.stanford.edu/extra-notes/hoeffding.pdf tybee island pizza deliveryWebJensen Inequality Theorem 1. Let fbe an integrable function de ned on [a;b] and let ˚be a continuous (this is not needed) convex function de ned at least on the set [m;M] where mis the int of fand Mis the sup of f. Then ˚(1 b a Z b a f) 1 b a Z b a ˚(f): Proof. We take the following de nition of a convex function. ˚is convex if for every ... tammy\u0027s picturesWebNov 12, 2024 · The Jensen inequality is a widely used tool in a multitude of fields, such as for example information theory and machine learning. It can be also used to derive other standard inequalities such as the inequality of arithmetic and geometric means or the Hölder inequality. In a probabilistic setting, the Jensen inequality describes the … tammy\u0027s on the aveWebExercises 1. Let A and B b e arbitrary ev en ts. Pro v e Bo ole's ine quality: P (\) P (B c)=1 … tybee island parking services