Central Limit Theorem

Lots of Dice Rolls

We know that dice rolls follow a uniform distribution over {1, 2, 3, 4, 5, 6}. However, if we take a large number of means of hundred dice rolls, we get a normal-ish distribution.

n <- 1e5 # number of times you want to take the mean
N <- 1e2 # number of dicerolls per mean

samps<- numeric(n)

for(i in 1:n)  samps[i] <- mean(sample(1:6, N, replace =T))

plot(density(samps))

variance <- var(sample(1:6, 1e4, T))/(N)
# to get true distribution, we can either estimate the plot 
#lines(density(rnorm(1e8, 3.5, sqrt(variance))) , col = "red") t

# or approximate it (faster) 
x <- 200:500 /100
lines(x, dnorm(x, 3.5, sqrt(variance)), col = "red")
legend("topleft",legend = c("True", "Expected"), lty = c(1,1), col = c("black","red") )

variance 

[1] 0.02946548

We can cross check by calculating the variance by hand, which equals \(\sigma^2/N\) for samples of size N. It comes out as

35/1200

[1] 0.02916667

Close enough.