Does $2\int_0^\infty P[X \ge t] t dt $ have meaning for 'general' random variables?

Question

Does \begin{align} 2\int_0^\infty P[X \ge t] t dt \end{align} have any meaning for a general random variable?

It is well known that for positive random variables \begin{align} 2\int_0^\infty P[X \ge t] t dt= E[X^2] \end{align}

If $X$ is a symmetric random variable then \begin{align} P[|X| \ge t] =2 P[ X\ge t] \end{align}

\begin{align} 2\int_0^\infty P[X \ge t] t dt= \int_0^\infty P[ |X| \ge t] t dt= \frac{1}{2}E[X^2] \end{align}

What about a more general set case of $X$?

Answer 1

$\mathbb{E}[\max(X,0)^2]$, if you consider that "meaningful". More generally, $$ \int_0^\infty \mathbb{P}(X \geq t)\;t^n\;\mathrm{d}t = \frac{1}{n+1}\mathbb{E}\left[\max(X,0)^{n+1}\right],$$ assuming $\mathbb{P}(X \geq t) \in o(1/t^{n+1})$ as $t \to \infty$.