I was wondering where there is a general formula to relate the expected value of a continuous random variable as a function of the quantiles of the same r.v.
The expected value of r.v. $X$ is defined as:
$E(X) = \int x dF_X(x) $
and quantiles are defined as :
$Q^p_X = \{x : F_X(x) = p \} =F_X^{-1}(p) $ for $p\in(0,1)$.
Is there for instance a function function $G$ such that: $E(X) = \int_{p\in(0,1)} G(Q^p_X) dp $
The inverse (right inverse in discrete case) of the cumulative distribution function $F(x)$ is called the quantile function, often denoted $Q(p)=F^{-1}(p)$. The expectation $\mu$ can be given in terms of the quantile function (when the expectation exists ...) as $$ \mu=\int_0^1 Q(p)\; dp $$ For the continuous case, this can be showed via a simple substitution in the integral: Write $$ \mu = \int x f(x) \; dx $$ and then $p=F(x)$ via implicit differentiation leads to $dp = f(x) \; dx$: $$ \mu = \int x \; dp = \int_0^1 Q(p) \; dp $$ We got $x=Q(p)$ from $p=F(x)$ by applying $Q$ on both sides.
For the general case, we can interpret $E(X) = \int x dF_X(x)$ as a Riemann–Stieltjes integral which, when the range of $X$ is the finite interval $[a,b]$, is defined as a limit of approximating sums of the form, for partitions $a=x_0<x_1<\dotsm<x_n=b$, $$\sum_{i=0}^{n-1} x_i^* \left[F(x_{i+1})-F(x_i)\right] $$ Passing to the quantile function $F^*$ (read the $^*$ as indicating a generalized inverse), and using $p_i=F(x_i)$, $p_i^*=F(x_i^*)$ this goes over to the approximating sum $$ \sum_{i=0}^{n-1} F^*(p_i^*) \left[ p_{i+1} - p_i \right] $$ which are approximating sums for $$ \int_0^1 F^*(p) \; dp $$ Then the infinite range case is treated the same way as with Riemann integrals.
