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Jensen's inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. Before embarking on these mathematical derivations, however, it is worth analyzing an intuitive graphical argument based on the probabilistic case where X is a real number (see figure).
There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means.
Since the square root is a strictly concave function, it follows from Jensen's inequality that the square root of the sample variance is an underestimate. The use of n − 1 instead of n in the formula for the sample variance is known as Bessel's correction , which corrects the bias in the estimation of the population variance, and some, but ...
Download as PDF; Printable version; ... In mathematics, Jensen's theorem may refer to: Johan Jensen's inequality for convex functions; Johan Jensen's formula in ...
Note that the operator ... for example. Löwner–Heinz theorem ... satisfies Jensen's Operator Inequality if the following holds ...
Example of samples from two populations with the same mean but different standard deviations. Red population has mean 100 and SD 10; blue population has mean 100 and SD 50. A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean.
Instead, he was a successful engineer for the Copenhagen Telephone Company between 1881 and 1924, and became head of the technical department in 1890. All his mathematics research was carried out in his spare time. Jensen is mostly renowned for his famous inequality, Jensen's inequality. In 1915, Jensen also proved Jensen's formula in complex ...
Hölder's inequality; Jackson's inequality; Jensen's inequality; Khabibullin's conjecture on integral inequalities; Kantorovich inequality; Karamata's inequality; Korn's inequality; Ladyzhenskaya's inequality; Landau–Kolmogorov inequality; Lebedev–Milin inequality; Lieb–Thirring inequality; Littlewood's 4/3 inequality; Markov brothers ...