optimization perspective on Markov's inequality

Here’s an intuitive explanation of Markov’s inequality. Suppose we fix the probability $p_{\alpha }=\mathbb{P}(X\ge \alpha )$ and ask the question: How small can we make $\mathbb{E}[X]$ subject to the constraint that $X\ge 0$ a.s.?

That is, we want to solve

$$\begin{array}{rl}\underset{P}{\min }& {\mathbb{E}}_{X\sim P}[X]\\ \text{s.t}& p_{\alpha }={\mathbb{P}}_P(X\ge \alpha )\\ & X\ge 0.\end{array}$$

This is easy. If we could, we’d put all of $X$‘s mass at 0, which would minimize $\mathbb{E}[X]$ subject to $X\ge 0$. But we have to move at least some of the mass to $\alpha$ or beyond. So we’ll put mass $p$ at $\alpha$, and mass $1-p$ at 0. It should be clear that any other alteration would simply increase $\mathbb{E}[X]$. With these choices, we have $\mathbb{E}[X]=\alpha p_{\alpha }$. Since this minimized $\mathbb{E}[X]$, we have $\mathbb{E}[X]\ge \alpha p_{\alpha }$ for all other $X$ which is precisely Markov’s inequality.