Jensen's inequality

Jensen's inequality is an inequality that characterizes convex functions. In the context of probability theory, it states that, given a function ff of a Random variable XX, if ff is convex, then

E[f(X)]≥f(E[X])E[f(X)]\geq f(E[X])

where EE is the Expected value.

A function f(x)f(x) is said to be convex if

∀x,y∈Rand∀α∈[0,1],f(αx+(1−α)y)≤αf(x)+(1−α)f(y)\forall x,y\in \mathbb{R} \quad\text{and}\quad \forall \alpha \in[0,1] ,\quad f(\alpha x+(1-\alpha)y)\leq \alpha f(x)+(1-\alpha)f(y)