Jensen inequality exercises
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 WebJensen-like inequalities are introduced, as well as a generalisation of a recent improvement to Jensen's inequality. Some of their applications are proposed: extensions of Lyapunov's inequality and inferential problems. After discussing upper and lower Markov's inequalities, Cantelli-like inequalities are proven with different degrees of ...
Jensen inequality exercises
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WebIn mathematics, Jensen's inequality, named after the Danish mathematician Johan … WebExercises 25 Chapter II. Topology § 1. Category 27 § 2. Baire property 30 § 3. Borel sets 32 § 4. The space 3 35 § 5. Analytic sets 38 § 6. Operation A 41 ... Jensen inequality 181 § 2. Jensen-Steffensen inequalities 184 § 3. Inequalities for means 189 § 4. Hardy-Littlewood-Polya majorization principle 192 § 5. Lim's inequality 194
WebJensen's inequality is an inequality involving convexity of a function. We first make the … WebJul 31, 2024 · A Gentle Introduction to Jensen’s Inequality. It is common in statistics and …
WebApplication of Convex Functions and Jensen Inequality Hua Zhonga, Xuemei Yangb, WuSheng Wangc. School of Mathematics and Statistics, Hechi University, Guangxi, Yizhou 546300, P. R. China ... Drilling of mathematical analysis exercises, Science Press, Beijing, 2006. [3] J. Kuang, Common inequality Fourth Edition, Shandong science and Technology ... 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 ...
Webt. Jensen’s inequality says that f( 1x 1 + 2x 2 + + nx n) 1f(x 1) + 2f(x 2) + + nf(x n): When x …
http://www.probability.net/jensen.pdf east west shrine game ticketsWeb6.2.5 Jensen's Inequality. Remember that variance of every random variable X is a positive value, i.e., Var(X) = EX2 − (EX)2 ≥ 0. Thus, EX2 ≥ (EX)2. If we define g(x) = x2, we can write the above inequality as E[g(X)] ≥ g(E[X]). The function g(x) = x2 is an example of convex function. Jensen's inequality states that, for any convex ... cummings michaelWeba fand integrate the inequality ˚(f(x)) (f(x) x 0) + ˚(x 0): We get Z ˚(f) (x 0 x 0)(b a) + (b a)˚(x … east west shrine game tvWebTheorem 4 Jensen’s Inequality (Integral Version) Let (X; ) be a measure space with (X) = 1. Let ’: (a;b) !R be a convex ... as an exercise to the reader. Convexity, Inequalities, and Norms 9 Applying the same reasoning using the integral version of Jensen’s inequality gives p q ) Z X fpd 1=p X fqd 1=q cummings middle school brownsville shootinghttp://sepwww.stanford.edu/data/media/public/sep/jon/jensen.pdf east-west shrine game vs senior bowlWebJensen'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 cummings middle school memphisWebExercises 1. Let A and B b e arbitrary ev en ts. Pro v e Bo ole's ine quality: P (\) P (B c)=1 … cummings mills