Here is a neat little relation and I am wondering if/how it generalizes.
The complementary cumulative distribution functions of a random variable, $X>0$, with density, $\rho(x)$, is $$C(x)=\int_x^\infty \rho(x')\mbox{d}x',$$ which can be rewritten as an average over $\rho(x)$ of an indicator function, $$C(x)=\int_0^\infty \Theta(x-x')\rho(x')\mbox{d}x'=\left[\Theta(x-x')\right]_{\rho}.$$ A natural question regards what other indicator-like functions can be averaged to obtain $C(x)$. This will depend on the choice of $\rho(x)$.
For exponential densities, $\rho(x)=\lambda e^{-\lambda x}$, $C(x)$ can written as the average of any of the $q$-ordered family of functions, $f_q(x)=\frac{1}{q!}x^q$, for $q=0,1,2,\dots$, $$C(x)=\left[f_q(\lambda(x-x'))\Theta(x-x')\right]_{\rho},$$ demonstrated by applying integration by parts $q$ times. Intuitively, this is possible because the mass lost for $x<x'-\lambda^{-1}$ is compensated by the mass added over $x>x'-\lambda^{-1}$, possible because $x^q$ is increasing. The $1/q!$ pre-factor makes this compensation exact.
So, can you find other classes of densities and functions that give this result?
Any response welcome. Thanks!