10
$\begingroup$

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 $

$\endgroup$
15
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Can you have a look at this question please? I think your insights might be helpful. $\endgroup$ – luchonacho Sep 4 '18 at 14:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.