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.

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