Jensen's inequality proof
WebFor example, in the proof of H older’s inequality below, we use gde ned on a set with just two points, assigned weights (measures) 1 p and 1 q with 1 p + q = 1. In that case the statement of Jensen’s inequality becomes [3.6] Theorem: (Jensen) Let gbe an R-valued function on the two-point set f0;1gwith a WebJan 13, 2024 · I was interested to see a proof for Jensen's inequality for the following variant: Let X be a discrete random variable with finite expected value and let h: R → R be a convex function. then: h ( E [ X]) ≤ E [ h ( X)] Please note, I'm interested in a proof for this variant with a discrete random variable.
Jensen's inequality proof
Did you know?
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder … See more The classical form of Jensen's inequality involves several numbers and weights. The inequality can be stated quite generally using either the language of measure theory or (equivalently) probability. In the … See more Form involving a probability density function Suppose Ω is a measurable subset of the real line and f(x) is a non-negative function such that $${\displaystyle \int _{-\infty }^{\infty }f(x)\,dx=1.}$$ See more • Jensen's Operator Inequality of Hansen and Pedersen. • "Jensen inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994] See more Jensen's inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be … See more • Karamata's inequality for a more general inequality • Popoviciu's inequality • Law of averages See more WebHoeffding’s inequality is a powerful technique—perhaps the most important inequality in learning theory—for bounding the probability that sums of bounded random variables are too large or too small. We will state the inequality, and then we will prove a weakened version of it based on our moment generating function calculations earlier.
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 … WebJensen’s Inequality is a statement about the relative size of the expectation of a function compared with the function over that expectation (with respect to some random variable). …
WebThis executive home of 5+Br/5Ba, 4900esf home has a quiet cul-de-sac location in the ever-popular and sought-after Encinitas Ranch. Immaculately maintained wood floors, brand … WebJensen’s Inequality is a statement about the relative size of the expectation of a function compared with the function over that expectation (with respect to some random variable). To understand the mechanics, I first define convex functions and then walkthrough the logic behind the inequality itself. 2.1.1 Convex functions
http://ele-math.com/static/pdf/books/17689-MIA19.pdf
WebProof by Convexity. We note that the function is strictly concave. Then by Jensen's Inequality, with equality if and only if all the are equal. Since is a strictly increasing function, it then follows that with equality if and only if all the are equal, as desired. Alternate Proof by Convexity. This proof is due to G. Pólya. rogue deck of many thingsWebProof We proceed by induction on n, the number of weights. If n= 1 then equality holds and the inequality is trivially true. Let us suppose, inductively, that Jensen’s inequality holds for n= k 1. We seek to prove the inequality when n= k. Let us then suppose that w 1;w 2;:::w k be weights with w j 0 P k j=1 w j = 1 If w k = 1 then the ... our teachers their way to meetWebJensen’s Inequality: Let C Rdbe convex and suppose that X2C. Provided that all expectations are well-defined, the following hold. (1)The expectation EX2C (2)If f: C!R is convex then f(EX) Ef(X). If fis strictly convex and Xis not constant then the inequality is strict. (3)If f: C!R is concave then f(EX) Ef(X). If fis strictly concave and Xis our teachers are the bestWebJan 1, 2024 · Purpose: This package contains the forms needed to document and request reimbursement for overnight travel.It includes the Travel Authorization Form, Travel … our teacher summaryWebSep 13, 2024 · The 80th percentile earned $68,000 in 2024, more than twice as much as the median worker in North Carolina. The top 20% of workers—those earning more than … our teacher told us that if we don\\u0027tWebJensen'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 our teacher was very proudWebSep 1, 2024 · Jensen’s inequality. We are now ready to formulate and prove Jensen’s inequality. It is an assertion about how convex functions interact with expected values of random variables, and we will formulate it on an abstract measure space $(\Omega, \Sigma, \P)$ where $\Omega$ is a set, $\Sigma$ is a $\sigma$-algebra of subsets of $\Sigma$, … our teacher was by the funny story