nrep <- 100000
out <- rep(0,nrep)
for (i in 1:nrep)
{
   s <- 0
   n <- 0
   while (s<=1) {
      n <- n+1
      x <- runif(1)
      s <- s+x
   }
   out[i] <- n
}
table(out)

xm <- mean(out)
xm

se <- sqrt(var(out)/nrep)
cil <- xm - 1.96*se
cih <- xm + 1.96*se
cat(xm,cil,cih,(cih-cil),"\n")