pdf/cdf integral

Question

I was wondering how do you compute the integral of pdf times some variable, that is divided by its cdf; to be more specific, our problem is

$\int_0^x\frac{2tg(t)}{G(t)}dt$

Where $g(t)$ is a PDF of i.i.d variables and $G(t)$ is its CDF.

The multiplication by $t$ makes this problem very hard to solve for us, but hopefully does someone know the answer!

Thanks in advance!

Answer 1

Note that $g(t) = G'(t)$. Now, you may write the integral as $$\int_0^x\frac{2tg(t)}{G(t)}dt =2\int_0^xt\frac{G'(t)}{G(t)}dt$$ Further, note that $\frac{d\ln{G(t)}}{dt}= \frac{G'(t)}{G(t)}$.

So, you may use partial integration. But whether you can integrate the arising logarithmic integral depends heavily on $G$.