## Problem 1  

k <- 1
p <- 0.5
reps <- 1000

notDone <- TRUE

while ( notDone )
{
 #Generate k*reps draws from the modified geometric distribution
 draws <- rgeom(k*reps, p)+1
 
 X <-array(draws, c(k,reps))
 Ak <- apply( X, 2, mean)

 # Expected value of our geometric sample mean is 1/p
 EAk <- 1/p

 # Std Dev of our geometric RVs Xi is sqrt( (1-p)/p^2 )
 sdk <- sqrt((1-p)/(p^2) )

 Z <- sqrt(k) *(Ak-EAk)/sdk
 diff80 <- abs(quantile(Z, 0.8) - 0.85)
 diff90 <- abs(quantile(Z, 0.9) - 1.28)
 diff95 <- abs(quantile(Z, 0.95) - 1.65)

 if( diff80 < .05 && diff90 < .05 && diff95 < .05 )
 {
   notDone <- FALSE
 }else {k <- k+1}
}
print(k)
plot(density(Z), main="Distribution of Xbar", ylab = "p(x)", xlab="x")