If X is a continuous random variable and E(X) exists, does the limit as x→∞ of x[1 - F(x)] = 0?(adsbygoogle = window.adsbygoogle || []).push({});

I encountered this, but so far I have neither been able to prove this, nor find a counterexample. I have tried the mathematical definition of the limit, l'Hopital's rule, integration by parts, a double integral (through expectation), and various proof scribbles, but so far, nothing has worked. Can anyone help me with this?

EDIT: In this case, the function F is the CDF of X.

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# Expectation: Is this proposition true or false?

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