Jensen's inequality Lemma. Let f be a convex function, and let X be a random variable.Then,f(E[Ⅺ)≤Ef(X) 700 f(E[X)=/∑k,P{X=k} k=-00 ∑f(k)Pr{X=k k Convexity lemma(generalized) c 2001 by Charles E Leiserson Introduction to Agorithms Day17L9.14© 2001 by Charles E. Leiserson Introduction to Algorithms Day 17 L9.14 Jensen’s inequality ∑ ∑ ∞ =−∞ ∞ =−∞ ≤ ⋅ =   = ⋅ = k k f k X k f E X f k X k ( ) Pr{ } ( [ ]) Pr{ } Proof. Convexity lemma (generalized). Lemma. Let f be a convex function, and let X be a random variable. Then, f(E[X]) ≤ E[f(X)]