Does $P(X>0)\geq\mathbb{E}[X]^2/\mathbb{E}[X^2]$ hold for a non-negative random variable $X$?

Question

Is there a way to see if the inequality $$P(X>0)\geq\mathbb{E}[X]^2/\mathbb{E}[X^2]$$ is generally true for a random variable $X\geq0$ in $\mathbb{R}$?

Answer 1

Yes, it's true by Cauchy-Schwarz inequality:

$$\left(\int_{\Omega}X d\mu\right)^2=\left(\int_{\Omega}X \chi_{[0,\infty)} d\mu\right)^2\leq\left(\int_{\Omega}X^2 d\mu\right)\left(\int_{\Omega}\chi_{[0,\infty)}^2 d\mu \right)=\\=\left(\int_{\Omega}X^2 d\mu\right)\left(\int_{\Omega}\chi_{[0,\infty)} d\mu \right)=\\=\left(\int_{\Omega}X^2 d\mu\right)P(X>0)$$