Finding second moment given $P(X>t)$

Question

I know that for nonnegative continuous R.V, $E[X]=\int_0^\infty P(X>t)dt$. Is there a formula for $E[X^2]$ when we only have $P(X>t)$?

Answer 1

Using integration by parts one may show: $$EX^2=\int_0^{\infty}2xP(X>x)dx$$

the tails should go to zero fast enough though :-)

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Addendum

\begin{align} \int_0^{\infty}2xP(X>x)dx&=\int_0^{\infty}2x\left(1-F(x)\right)dx\\ &=x^2\left(1-F(x)\right)|_0^{\infty}+\int_0^{\infty}x^2f(x)dx \end{align} Note now that we need $\lim_{x\to\infty}x^2(1-F(x))=0$, i.e. the tails should vanish faster than $x^2$.