Discussion 7

Andy Carson

5.1 Problem 2

Let n be a positive integer. Let S be the set of integers between 1 and n. Consider the following process: We remove a number from S at random and write it down. We repeat this until S is empty. The result is a permutation of the integers from 1 to n. Let X denote this permutation. Is X uniformly distributed?

Answer: No, X is not uniformly distributed.  And how it is distributed changes a lot depending on the size of n.  In general, there are gaps in the distribution for values that cannot be formed (or are much less likely to be formed).  And when one gets to values above 9, since the digit "1" is appearing more often, that tends to show up more as the leading digit. Similarly for going above 20 or 30 for values starting with "2" or "3".  Here are some examples:

library(stringr)

for(j in c(5,10,15,20,25,30)) {
  n<-j
  s<-1:n
  trials<-100000
  simulations<-rep(NA,trials)
  
  
  for(i in 1:trials){
    simulations[i]<-as.numeric(str_c(sample(s,n, replace = FALSE),collapse = ""))
  } #i
  
  
  hist(simulations, breaks = 100, main = j)
  
  
}#j