order, order!

A very standard (one-line) question on X validated, namely whether min(X,Y) could enjoy a finite mean when both X and Y had infinite means [the answer is yes, possibly!] brought a lot of traffic, including an incorrect answer and bringing it to be one of the “Hot Network Questions“, for no clear reason. Beside my half-Cauchy example, some answers pointed out the connection between mean and cdf, as integrated cdf on the negative half-line and integrated complement cdf on the positive half-line, and between mean and quantile function, as

$\mathbb E[T(X)]=\int_0^1 T(Q_X(u))\text{d}u$

since it nicely expands to

$\mathbb E[T(X_{(k)})]=\int_0^1 \frac{u^{k-1}(1-u)^{n-k-1}}{B(k,n-k)}T(Q_X(u))\text{d}u$

but I remain bemused by the excitement..! (Including the many answers and the lack of involvement of the OP.)

This entry was posted on June 9, 2020 at 12:20 am and is filed under Books, pictures, Statistics, University life with tags Cauchy distribution, conditioning, John Bercow, Michel de L'Hôpital, order, order statistics, UK politics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Responses to “order, order!”

Gerard Letac Says:
June 9, 2020 at 4:05 pm

Un peu bébête:. Si $X$ est à valeurs entières alors
$E(X)=\sum_{n\geq 1} Pr(X\geq n)$ .
Si $X\sim Y$ avec indépendance et si
$Pr(X\geq n)=1/n$
alors
$Pr(\min(X,Y)\geq n)=1/n^2$ .

Reply
- xi'an Says:
  June 9, 2020 at 9:43 pm
  
  Effectivement, ça a le mérite de la simplicité!
  
  Reply