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Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. 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.
Hoeffding's inequality; Hoeffding's lemma; Jensen's inequality; Khintchine inequality; Kolmogorov's inequality; Kunita–Watanabe inequality; Le Cam's theorem; Lenglart's inequality; Marcinkiewicz–Zygmund inequality; Markov's inequality; McDiarmid's inequality; Paley–Zygmund inequality; Pinsker's inequality; Popoviciu's inequality on ...
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 ...
Jensen's inequality provides a simple lower bound on the moment-generating function: (), where is the mean of X. The moment-generating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable X .
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 ...
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.
The surge in Nvidia’s share price has significantly increased the wealth of its founder and CEO, Jensen Huang. According to Forbes, Huang now ranks as the 20th richest person in the world, with ...
Jensen's inequality - relates the value of a convex function of an integral to the integral of the convex function; John ellipsoid - E(K) associated to a convex body K in n-dimensional Euclidean space R n is the ellipsoid of maximal n-dimensional volume contained within K.