## overlap, overstreched

Posted in Books, Kids, R, Statistics with tags , , , , , , on June 15, 2020 by xi'an

An interesting challenge on The Riddler on the probability to see a random interval X’ing with all other random intervals when generating n intervals from Dirichlet D(1,1,1). As it happens the probability is always 2/3, whatever n>1, as shown by the R code below (where replicate cannot be replaced by rep!):

qro=function(n,T=1e3){
quo=function(n){
xyz=t(apply(matrix(runif(2*n),n),1,sort))
sum(xyz[,1]<min(xyz[,2])&xyz[,2]>max(xyz[,1]))<0}
mean(replicate(quo(n),T))}


and discussed more in details on X validated. As only a property on permutations and partitions. (The above picture is taken from this 2015 X validated post.)

## order, order!

Posted in Books, pictures, Statistics, University life with tags , , , , , , on June 9, 2020 by xi'an

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.)