fred.R

rm(list = ls())
set.seed(23456)

x <- 0:360
plot(x, - cos(x* pi /180), type = "l")
y <- rep(0, 361)
N <- 1e2

for (i in x) y[i + 1] <- rbinom(1, N, (1 + cos(i * pi /180))/2)
plot(x, - cos(x* pi /180), type = "l")
lines(x, 1 - 2 * y/N)

plot(x, sqrt(N) * ((1 - 2 * y/N) + cos(x* pi /180)))

N <- 1e4

for (i in x) y[i + 1] <- rbinom(1, N, (1 + cos(i * pi /180))/2)
plot(x, - cos(x* pi /180), type = "l")
lines(x, 1 - 2 * y/N)

plot(x, sqrt(N) * ((1 - 2 * y/N) + cos(x* pi /180)))

N <- 1e6

for (i in x) y[i + 1] <- rbinom(1, N, (1 + cos(i * pi /180))/2)
plot(x, - cos(x* pi /180), type = "l")
lines(x, 1 - 2 * y/N)

plot(x, sqrt(N) * ((1 - 2 * y/N) + cos(x* pi /180)))

p <- (1 + cos(x * pi /180))/2
plot(x, sqrt(N) * ((1 - 2 * y/N) + cos(x* pi /180)) / sqrt(p * (1 - p)), ylim = c(-8, 8))