m=3
n=2
rF = 6
R = NULL
k=1
for ( i.1 in 1:(m+n-2) )
{
  for ( i.2 in (i.1+1):(m+n-1))
  {
    for ( i.3 in (i.2+1):(m+n) )
    {
      R[k] = i.1+i.2+i.3
      k=k+1 
    }
  }
}

## Probability that R = rF
sum(R == 6)/length(R) 

## P(R <= 6 and R >= 12)
sum(R <= 6)/length(R) + sum(R >= 12)/length(R) 


m = 5
n = 7

X = rnorm(m)
Y = rnorm(n)

## Compute W
W = 0
for (i in 1:m)
{
  W = W + sum(Y < X[i])
}

## Compute W1=Ranksum(X) - m(m+1)/2
Z = c(X, Y)
R = rank(Z)
RX = R[1:m]
W1 = sum(RX) - m*(m+1)/2
 
c(W, W1)





## Compute the proportion of times
## we reject Ho, if it is actually true
m = 25
n = 15

mu = m*n/2
s = sqrt(m*n*(m+n+1)/12)

nrep = 1e4
WS = NULL

for (k in 1:nrep) 
{
  ## Generate Data under H0
  X = rexp(m)
  Y = rexp(n)

  Z = c(X, Y)
  R = rank(Z)
  RX = R[1:m]
  W = sum(RX) - m*(m+1)/2

  WS[k] = (W - mu) / s
}
## Reject using alpha = 0.05
mean(abs(WS) > qnorm(0.975))