How to prove that $\mathbb E(x^n)-\mathbb E(x)^n \ge 0; \quad n \in \mathbb Z_+.$

Question

How to prove that $$\mathbb E(X^n)-\mathbb E(X)^n \ge 0; \quad n \in \mathbb Z_+.$$

X is a R.V. in the range of $[0, 1]$

Any hints appreciated. Thanks.

Answer 1

For $f(x)=x^n,x\in[0,1]$ is convex function,so by the Jensen's inequality: \begin{equation} E(f(X))=E(X^n)\geq f(E(X))=E(X)^n \end{equation}