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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 ...
Jensen's inequality: If : is a convex function, then (()) (()). Conditional variance : Using the conditional expectation we can define, by analogy with the definition of the variance as the mean square deviation from the average, the conditional variance
If p is less than 1/2, the gambler loses money on average, and the gambler's fortune over time is a supermartingale. If p is greater than 1/2, the gambler wins money on average, and the gambler's fortune over time is a submartingale. A convex function of a martingale is a submartingale, by Jensen's inequality.
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Firstly, while the sample variance (using Bessel's correction) is an unbiased estimator of the population variance, its square root, the sample standard deviation, is a biased estimate of the population standard deviation; because the square root is a concave function, the bias is downward, by Jensen's inequality.