## 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.)

### 2 Responses to “order, order!”

1. Gerard Letac Says:

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
$Pr(\min(X,Y)\geq n)=1/n^2$.

• Effectivement, ça a le mérite de la simplicité!

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