Inequality of first three moments for positive random variable

Question

Let $X$ be a positive random variable (let's say finitely supported). Let $\mu_r = \mathbb{E} X^r$ be the $r$-th moment. Is it true that $$\mu_3 \geq \mu_1 \cdot \mu_2$$

Answer 1

Yes! An easy way to see this: If $X\ge0$ then $X$ and $X^2$ are positively correlated (draw a picture). Thus $$\operatorname{Cov}(X,X^2)\ge0$$ which means $$E(XX^2)-E(X)E(X^2)\ge0.$$

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EDIT: More rigorously, the general result is: If $h$ is a nondecreasing function, then $\operatorname{Cov}(X,h(X))\ge0$. [Here $X\ge0$ so the function $h(x):=x^2$ qualifies.]

Proof: By definition, $$\operatorname{Cov}(X,h(X)):=E(X-EX)(h(X)-Eh(X)).$$ Write $$(X-EX)(h(X)-Eh(X))=(X-EX)(h(X)-h(EX)) + (X-EX)(h(EX)-Eh(X)).\tag1 $$ The first term on the RHS of (1) is a nonnegative random variable, since $h$ is nondecreasing. The second term has expectation zero since $h(EX)$ and $Eh(X)$ are both constant.