Распознавание и корректура текста из книги в формате djvu: ----------- Страница 180 1. Jensen\'s inequality. A function u: R→R is called convex if for all real a there exists λ, depending on a, such that u(x) ≥ u(a)+ λ (x – a) for all x. (Draw a diagram to illustrate this definition ). Show that, if u is convex and X is a random variable with finite mean, then E(u(X)) ≥ u(E(X)). Страница 185 2. If Ø is a characteristic function, show that Re{1 – φ(t)} ≥ Re{1 – φ(2t)}, and deduce that 1 – | φ(2t)| ≤ 8{1 – | φ(t)|}.