## Check some properties of uniform random numbers as simulated by R.


## Generate a large number of uniform random variables.
nrep = 1e6
U = runif(nrep)

## Calculate the proportion of draws in each decile, e.g. [0,0.1),
## [0.1,0.2), ..., [0.9,1).
M = NULL
for (k in 1:10) {
  I = ((U >= (k-1)*0.1) & (U < k*0.1))
  M[k] = sum(I)
}


## A crude test of non-independence: estimate the probability that a
## value is greater than 0.9, given that the preceeding value was
## greater than 0.9.
QN = 0
QD = 0
for (i in 2:nrep) {
  if (U[i-1] > 0.9) { 
    QD = QD+1
    if (U[i] > 0.9) { QN = QN+1 }
  }
}
p_est1 = QN/QD

## A vectorized version of the previous code.
p_est2 = sum((U[2:nrep]>0.9) & (U[1:(nrep-1)]>0.9)) / sum(U[1:(nrep-1)]>0.9)