#
# EPage 374
#
# Written by:
# -- 
# John L. Weatherwax                2009-04-21
# 
# email: wax@alum.mit.edu
# 
# Please send comments and especially bug reports to the
# above email address.
# 
#-----

library(DAAG)
library(MASS)
library(rpart)
library(randomForest)

set.seed(0)

ns = c(25,50,100,400,800)
empirically_repeated = c()
expected_repeated = c()
for( n in ns ){
  the_samps = sample( 1:n, size=n, replace=T )

  # Determine the number of repeated elements:
  #
  num_of_dups = sum( rle( sort(the_samps) )$lengths != 1  )

  expected_number_of_repeats = (1-n^(-1))^n

  empirically_repeated = c( empirically_repeated, num_of_dups/n )
  expected_repeated = c( expected_repeated, expected_number_of_repeats )
  print( sprintf( "sample size= %5d; empricial fraction of repeats= %5.3f; expected number of repeats= %5.3f", n, num_of_dups/n, expected_number_of_repeats ) ) 
}

# Part 6:
plot( empirically_repeated, expected_repeated, type='p', xlab="empirically observed fraction of repeats", ylab="expected fraction of repeats" )

# Part 7: 
ys = exp(-1) - (1-ns^(-1))^ns
plot( log(ns), ys )