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. 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
Jensen–Steffensen inequality for strongly convex functions
WebApr 15, 2024 · for any \(n\ge 1\).The Turán inequalities are also called the Newton’s inequalities [13, 14, 26].A polynomial is said to be log-concave if the sequence of its coefficients is log-concave. Boros and Moll [] introduced the notion of infinite log-concavity and conjectured that the sequence \(\{d_\ell (m)\}_{\ell =0}^m\) is infinitely log-concave, … WebJul 6, 2010 · In this chapter, we shall establish Jensen's inequality, the most fundamental of these inequalities, in various forms. A subset C of a real or complex vector space E is … pallet rack inspection training
A PROOFOF JENSEN’SINEQUALITY - Project Euclid
WebDec 24, 2024 · STA 711 Week 5 R L Wolpert Theorem 1 (Jensen’s Inequality) Let ϕ be a convex function on R and let X ∈ L1 be integrable. Then ϕ E[X]≤ E ϕ(X) One proof with a nice geometric feel relies on finding a tangent line to the graph of ϕ at the point µ = E[X].To start, note by convexity that for any a < b < c, ϕ(b) lies below the value at x = b of the linear … WebJensen’s inequality by taking the convex function to be the exponential function. The above proof specialized to this case is similar to the proof given in [1], though in this proof the property that the derivative of the natural logarithm is decreasing was used instead. The statement of Jensen’s inequality for integrals is taken from [6]. 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 pallet rack inspection new jersey