Jensen's inequality
The expression of variance \[ Var(X)=E[(X-EX)^2]=EX^2-(EX)^2\geq 0 \] Implies that \[ EX^2\geq (EX)^2 \] Here \(X^2\) is an example of the convex function. The definition of convex function is
A twice-differentiable function \(g:I\rightarrow \mathbb{R}\) is convex if and only if \(g''(x)\geq 0\) for all \(x\in I\)
Below is a typical example of the convex function. The function is convex if the line segment between two points from the curve lies above the curve. On the other hand, if the line segment aloways lies below the curve, then the function is said to be concave.
The Jensen's inequality states that for any convex function \(g\), we have \(E[g(X)]\geq g(E(X))\). To be specific.
If \(g(x)\) is a convex function on \(R_X\), and \(E[g(X)]\) and \(g[E(X)]\) are finite, then \[ E[g(X)]\geq g[E(X)] \]