F(1-F)
When answering an X validated question about the covariance between a random variable X and its cdf transform F(X), I realised that it was half the integral of the function
x → F(x)(1-F(x))
when X is centred. It is not surprising in the least to see the cdf appearing for this second order expectation, since it can similarly be used to represent first order expectations (as exploited by nested sampling). But it is easy to be confused by the fact that F(X) is usually a Uniform (0,1) variate hence distribution-free, until one sees it remains positively correlated with X, or by the apparent lack of scale or by the symmetry, until one realises this is not the case. (The associated correlation is scale-free.)
March 9, 2022 at 9:11 am
Jolie egalite qui merite que ton nom passe a la posterite!
Autre demonstration, qui remplace l’integration par parties par Fubini. Donc, si E(X)=0 et si X,Y sont iid on a (on suppose F continue pour simplifier)
March 9, 2022 at 2:08 pm
Envoie moi la formule par l’E-poste!