I have no idea how to proceed with proving this. If $X$ is a continuous random variable, $P(X > 0) = 1$, $E(X)$ is defined and $F(x)$ is the CDF, then prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$
2026-03-25 09:26:15.1774430775
Prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$
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Note that for $x>0$ $$ xI(X> x)\leq XI(X>x) $$ where $I$ is the indicator function. By taking expectations we find that $$ 0\leq xP(X>x)\leq EXI(X>x)\to 0 $$ as $x\to \infty$ by the dominated convergence theorem.