Can anyone help me to show that for a general random variable X with a fixed distribution to model the time interval between the independent events. The expected residual time, which is the time one has to wait for the event happen if she arrive at a random time is $frac{E(X^2)}{2E(X)}$.

I am expecting a complete proof for it. I already know the relationship between the expected residual time with the survival function S(x) but still not sure how to derive this.