# Upper Bound for 2nd Raw Moment of Positive Random Variable

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?

• 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.
– whuber
Commented Feb 15, 2021 at 13:44