Jensen inequality example
WebJensen’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 http://sepwww.stanford.edu/sep/prof/pvi/jen/paper_html/node2.html
Jensen inequality example
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WebApr 12, 2024 · For example, an AI-powered chatbot that is designed to help people find jobs could be more likely to recommend jobs that are traditionally held by men to men and jobs that are traditionally held ... WebFeb 23, 2024 · In studying the Jensen inequality, the following example is presented: Example 10.1.6 (Bias of sample standard deviation). Let X 1, …, X n be i.i.d. random variables with variance σ 2. Recall from Theorem 6.3.4 that the sample variance S n 2 is unbiased for estimating σ 2. That is, E ( S n 2) = σ 2.
Webneed to use a form of Jensen’s Inequality. Jensen’s Inequality says that if f is a concave down ... For example, ‘A’ is 1000001 and ‘Q’ is 1010001. On the other hand, with Morse Code, each letter is stored using a sequence of dots and dashes 5 (except for the ‘space’ character, which is stored with a space). For example, ‘A ... WebJensen’s inequality can be used to deduce inequalities such as the arithmetic-geometric mean inequality and Hölder’s inequality. Inequalities play an important role in almost all branches of mathematics as well as in other areas of science. ... The examples of energy optimal trajectories refer to the lines of the Bucharest subway. Full article
WebMar 24, 2024 · (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 … WebThis is an example of an exponential tail inequality. Comparing with Chebyshev’s inequality we should observe two things: 1. Both inequalities say roughly that the deviation of the average from the expected value goes down as 1= p n. 2. However, the Gaussian tail bound says if the random variables are actually Gaussian
WebOne way would be to apply the finite Jensen's inequality φ( ∑ aixi ∑ aj) ≤ ∑ aiφ(xi) ∑ aj to each Riemann sum. The finite inequality is itself easily proved by induction on the number …
http://cs229.stanford.edu/extra-notes/hoeffding.pdf fiftys couchWebNov 12, 2024 · The Jensen inequality is a widely used tool in a multitude of fields, such as for example information theory and machine learning. It can be also used to derive other standard inequalities such as the inequality of arithmetic and geometric means or the Hölder inequality. grimshaw mortonWebApr 29, 2024 · Jensen's inequality for indefinite integrals. In many sources, for example here, we can find the following generalisation of Jensen's inequality. In real analysis, we may require an estimate on φ ( ∫ a b f ( x) d x), where a, b ∈ R and f: [ a, b] → R is a non-negative Lebesgue-integrable function. In this case, the Lebesgue measure of ... fifty seasons easter buffetWeb• Jensen’s inequality says nothing about functions fthat are neither convex nor concave, while the graph convex hull bounds hold for arbitrary functions. • While Jensen’s inequality requires a convex domain Kof f, the graph convex hull bounds have no restrictions on the domain it may even be disconnected, cf.Example 3.9and Figure 3.1. fifty search divingSuppose 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 inequality? The natural logarithm is a strictly concave function because its second derivativeis strictly negative on its domain of definition. … 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 … 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 inequality: If the function is concave, then If the … See more If you like this page, StatLect has other pages on probabilistic inequalities: 1. Markov's inequality; 2. Chebyshev's inequality. 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 … See more fiftys diner north bay ontWebJensen's inequality is an inequality involving convexity of a function. We first make the following definitions: A function is convex on an interval I I if the segment between any … fifty scariest animals in the worldWebn Jensen’s inequality states: f(w 1x 1 +w 2x 2 +:::w nx n) w 1f(x 1)+w 2f(x 2)+:::+w nf(x n) Proof 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 ... fifty season 3