Jensen inequality exercises
http://www.sef.hku.hk/~wsuen/teaching/micro/jensen.pdf WebInequality involving Jensen (Rudin's exercise) Exercise (Rudin, R&CA, no. 3.25). Suppose μ …
Jensen inequality exercises
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WebUsing Jensen´s inequality to explain the role of regular….pdf. 2024-07-05上传. Using Jensen´s inequality to explain the role of regular… WebJensen 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 ...
WebDec 4, 2024 · So in class we solved the following exercise: state Jensen’s inequality for a convex function and use it to show that for a finite set of real numbers the geometric mean is less than or equal to the arithmetic mean. Unfortunately I lost my notes and cannot recall the proof at all. Jensen's inequalityis in Royden's book as the following:
Webthe inequality goes, and remembering a picture like this is a good way to quickly gure out the answer. Remark. Recall that f is [strictly] concave if and only if f is [strictly] convex (i.e., f00(x) 0 or H 0). Jensen’s inequality also holds for concave functions f, but with the direction of all the inequalities reversed (E[f(X)] f(EX), etc.). Below you can find some exercises with explained solutions. Exercise 1 Let be a random variable having finite mean and variance . Use Jensen's inequality to find a bound on the expected value of . Solution Exercise 2 Let be a positive integrable random variable. Find a bound on the mean of . Solution How to … See more Jensen's inequality applies to convex and concave functions. The properties of these functions that are relevant for understanding the proof of the inequality are: 1. the tangents of a convex function lie entirely below its … See more The following is a formal statement of the inequality. If the function is strictly convex and is not almost surelyconstant, then we have a strict … See more Jensen's inequality has many applications in statistics. Two important ones are in the proofs of: 1. the non-negativity of the Kullback-Leibler divergence; 2. the information inequality … See more Suppose that a strictly positive random variable has expected valueand it is not constant with probability one. What can we say about the expected value of , by using Jensen's … See more
WebExercises 1. Let A and B b e arbitrary ev en ts. Pro v e Bo ole's ine quality: P (\) P (B c)=1 …
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 how to lose my double chinWebJensen’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: how to lose my voice without screamingWebExercise 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) journal of alzheimer\u0027s disease \u0026 parkinsonismWebQuestion: Exercise 47 Check numerically that Jensen's inequality holds for the convex function \( f(x)=-\sqrt{x}, x \in \mathbb{R}_{+} \). This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. journal of alzheimer\u0027s disease 影响因子WebQuestion: Exercise 47 Check numerically that Jensen's inequality holds for the convex … how to lose my stomach fastWebTutorial 8: Jensen inequality 12 Exercise 10. Let (Ω,T) be a compact, non-empty … how to lose my love handleshttp://www.probability.net/jensen.pdf journal of alzheimer\u0027s research