I would like to know if my calculations are right. I tried to show that:
$\int_{-\infty}^{\infty} K_i(t)dt = E\xi_i^2$
where
$K_i(t) = E\{\xi_i (1_{0\leq t \leq \xi_i} - 1_{\xi_i\leq t < 0}) \}dt$
I got:
$\int_{-\infty}^{\infty} E\{\xi_i (1_{0\leq t \leq \xi_i} - 1_{\xi_i\leq t < 0}) \}dt=$ $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} t( 1_{0\leq t \leq x} - 1_{x\leq t < 0})dF_{\xi_i}(x)dt=$ $\int_{-\infty}^{\infty}[\int_{0}^{x} tdt -\int_{x}^{0} tdt ]dF_{\xi_i}(x)=$ $\int_{-\infty}^{\infty}x^2dF_{\xi_i}(x)=$ $E\xi_i^2$
Where I used Fubini once.
Thanks for any help.