# Central Limit Theorem
set.seed(532)

#The CLT is a very powerful result that says the sample mean of
# i.i.d. RVs is approximately normal if take large enough samples

# Simulation using standardized RV for Bernoulli seq (p.378)
# Should converge to standard normal RV
p <- 1/2
cltsequence <- function(n) (mean(rbinom(n, 1, p))-p) * (2*sqrt(n))

n <- 1000
simlist <- replicate(10000, cltsequence(n))
hist(simlist, prob = TRUE,xlab="Z-score",main="Histogram of CLT simulation")
curve(dnorm(x), -4, 4, add = TRUE) 

# Sum of 6 dice is approx normal (p. 383)
nsim <- 100000
ndice <- 6
mu <- ndice*3.5
sigma <- sqrt(sum(((1:6)-3.5)^2)*1/6*ndice)
simlist <- replicate(nsim, (sum(sample(1:6, ndice, rep=TRUE))-mu)/sigma)
hist(simlist[abs(simlist)<4],breaks=seq(from=-4,to=4,by=.25),freq=FALSE,xlab="Z-score",main="Histogram of CLT simulation")
curve(dnorm(x), -4, 4, add = TRUE) 

# Test whether our simulation of the normal distribution follows the 
# 68-95-99.7 rule for standard normal RV (mean 0, SD 1):
sum(-1 <=simlist & simlist <= 1)/n # proportion within one SD
sum(-2 <=simlist & simlist <= 2)/n # proportion within two SD
sum(-3 <=simlist & simlist <= 3)/n # proportion within three SD