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Let $X$ be a random variable with support $(0,\infty)$. All I know about $X$ is the support, finite higher moments, and $\mathbb{E}(X)=\mu$. I am trying to come up with a more tractable upper bound on $\mathbb{E}(X^2)$. So far I have

$$\mathbb{E}(X^2)=\int_0^\infty 2x P(X\geq x) dx$$ $$\leq\int_0^\infty 2x \underset{t>0}{\text{ min }} e^{-tx}M_X(t) dx$$

$$=\underset{t>0}{\text{ min }}\frac{2\times M_X(t)}{t^2}.$$

I do not know what $M_X(t)$ is though, though I know it exists and the first raw moment. If $X$ were bounded, there are useful inequalities, but does anyone know how to do this minimization, or know an upper bound for $M_X(t)$ for random variables with positive support?

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    $\begingroup$ Conditional on knowing only the first raw moment, the second raw moment could be anything larger than the squared raw moment--and can even be infinite. Thus there's no useful upper bound. $\endgroup$
    – whuber
    Commented Feb 15, 2021 at 13:44

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