## estimate standard deviation of the maximum as an estimate of upper limit
n=20

sdhats1 = sdhats2 = NULL
origmaxs = NULL
for (i in 1:10000){
	X = runif(n)
	maxs1=maxs2=NULL
	for (b in 1:1000){
	 	maxs1 = c(maxs1,max(sample(X, n, replace = T)))
	 	maxs2 = c(maxs2,max(runif(n)*max(X)))

	}
	sdhats1 = c(sdhats1, sd(maxs1))
	sdhats2 = c(sdhats2, sd(maxs2))
	origmaxs = c(origmaxs, max(X))
}
summary(sdhats1)
summary(sdhats2)
sd(origmaxs)		
	
# true value: 
sqrt(n/( (n+1)^2 * (n+2)))