Jensen's inequality

In mathematics, Jensen's inequality relates the value of a convex function of an integral to the integral of the convex function. Formally, it states that when f is convex and y is a non-negative function of x fulfilling

then

For example,

In probability theory, Jensen's inequality is often written using expectations. If f is a convex function and X is a random variable, then

Jensen's inequality is also valid when applied to a sequence of positive numbers. Given nonnegative numbers

and weights

such that

as well as a convex function f, we have that