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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).
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 ...
Indeed, convex functions are exactly those that satisfies the hypothesis of Jensen's inequality. A first-order homogeneous function of two positive variables x {\displaystyle x} and y , {\displaystyle y,} (that is, a function satisfying f ( a x , a y ) = a f ( x , y ) {\displaystyle f(ax,ay)=af(x,y)} for all positive real a , x , y > 0 ...
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.
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
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.
Actual wealth gap explained Citing a myriad of causes -- from cheap credit to exploitative bank practices -- they've noted that the average family puts away less than 4 percent of its income.
In mathematics, Jensen's theorem may refer to: Johan Jensen's inequality for convex functions; Johan Jensen's formula in complex analysis;