I am stuck on this problem for some time now. It says if $X$ is non-negative random variable with density function $f$ and if $r(u)=\int_{u}^{\infty}f(t)dt$, then
$E(X)=\int_{0}^{\infty}P(X\geq u)du=\int_{0}^{\infty}r(u)du$.
What I have observed is $r(u)=1-F(u)$, where $F$ is cumulative distribution function of X.
This means $r'(u)=-F'(u)=-f(u)$.
So, $E(X)=\int_{0}^{\infty}xf(x)dx=\int_{0}^{\infty}-xr'(x)dx$. But this approach doesn't seem working. Any hint. Thanks.