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, [ 1 ] building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. [ 2 ]

Learn how Jensen's inequality is stated and proved. Learn how to use it through examples and solved exercises.

Jensen’s inequality Theorem (Jensen’s Inequality) Let X be a random variable, and let g : R →R be a convex function. Then E[g(X)] ≥g(E[X]). (1) Where does it come from? Var[X] = E[X2] −E[X]2. Since Var[X] ≥0 for any X, it follows that E[X2] | {z } =E[g(X)] ≥ E[X]2 | {z } =g(E[X]). (2) 2/8

Jensen’s Inequality Recall: The expected value of a random vector X= (X 1;:::;X d)tis defined by EX = (EX 1;:::;EX d)t2Rd Jensen’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).

Jensen's inequality is an inequality involving convexity of a function. We first make the following definitions: A function is convex on an interval \(I\) if the segment between any two points taken on its graph \((\)in \(I)\) lies above the graph. An example of a convex function is \(f(x)=x^2\).

Jensen's Inequality is an inequality discovered by Danish mathematician Johan Jensen in 1906.

4 天之前 · (1) If f is concave, then the inequality reverses, giving f(sum_(i=1)^np_ix_i)>=sum_(i=1)^np_if(x_i). (2) The special case of equal p_i=1/n with the concave function lnx gives ln(1/nsum_(i=1)^nx_i)>=1/nsum_(i=1)^nlnx_i, (3) which can be exponentiated to give the arithmetic mean-geometric mean inequality ...

Examples of Jensen inequalities The most familiar example of a Jensen inequality occurs when the weights are all equal to 1/N and the convex function is f(x) = x 2 .

2020年6月5日 · Equality holds if and only if either $ x ( t) = \textrm{ const } $ on $ D $ or if $ f $ is linear on $ x ( D) $. If $ f $ is a concave function, the inequality signs in (1) and (2) must be reversed. Inequality (1) was established by O. Hölder, and (2) by J.L. Jensen .

Jensen’s inequality is used to bound the “complicated” expression E[f(X)] by the simpler expression f(E[X]). Often these expression are actually very close to each other.