#--------------------------------------
#--- Random Walk
e <- rnorm(1000, 0, sd=1)
X <- cumsum(e)


plot(X, ylim=c(-4*sqrt(1000), 4*sqrt(1000)))


for (i in 1:100){

  e <- rnorm(1000, 0,1)
  X <- cumsum(e)
  lines(X)
  readline("---")

}



#--------------------------------------
#--- Scale to be Bro Mo

n = 1000

e <- rnorm(n, 0, sd=1/sqrt(n))
X <- cumsum(e)

plot((1:n)/n, X, ylim=c(-4,4))


Y <- 0
for (i in 1:1000){

  e <- rnorm(n, 0,1/sqrt(n))
  X <- cumsum(e)
  lines((1:n)/n, X)
  readline("---")
  Y[i] <- X[n]

}

var(Y)