Robert & Casella. Chpt 2, Ex. 2.3

An antiquated generator of the normal distribution is:

Generate $U_{1}, U_{2}, U_{3}, .... , U_{12} \sim \mathcal{U}[-1/2, 1/2]$
Set $Z=\sum_{i=1}^{12}U_{i}$

The argument being that the CLT normality is sufficiently accurate with 12 terms

(a) Show that $\mathbb{E}(Z)=0$ and $\sigma^{2}(Z)=1$

For any random variable $U_{i}\sim \mathcal{U}[-1/2, 1/2]$ it follows from the definition of expected value and variance of the uniform distribution that $\mathbb{E}(U_{i})=0$ and $\sigma^{2}(U_{i})=1/12$ . Since $U_{1}, U_{2}, U_{3}, .... , U_{12}$ are i.i.d. then we can do:

$\mathbb{E}(Z)=\sum_{i=1}^{12}\mathbb{E}(U_{i})=(12)(0)=0$

$\sigma^{2}(Z)= \sum_{i=1}^{12}\sigma^{2}(U_{i})=(12)(1/12)=1$

(b) Using histograms, compare this CLT-normal generator with the Box-Muller algorithm.

(c)Compare both generators with rnorm

Here, I generated $10^{5}$ samples from three different normal generators and compared them by estimated PDFs, histograms and Q-Q plots. The samples generated from CLT-normal generator lacked fitness to the tail of $N(0,1)$ while the Box-Muller transformation worked quite well compared with the samples generated from the R function rnorm

The pictures are kinda crappy as you can see, so here’s the R code for them in case you’d like to reproduce them:

#ex 2.3 #Box-Muller's algorithm Box.Muller<-function(n){
U1<-runif(n) U2<-runif(n) sqrt(-2*log(U1))*cos(2*pi*U2)
}
#CLT normal generator CLT.Norm<-function(n){


matrix(runif(n*12, -.5, .5), ncol=12) %*% rep(1,12)

}
set.seed(100) n<-1e5 rv.bm<-Box.Muller(n) rv.clt<-CLT.Norm(n) rv.rnorm<-rnorm(n)
#Histograms and Q-Q plots par(mfrow=c(2,3)) hist(rv.rnorm, 50, main="Histogram of rnorm") hist(rv.bm, 50, main="Histogram of Box-Muller generator") hist(rv.clt, 50, main="Histogram of CLT generator") qqplot(rv.bm, rv.clt, pch=".", main="Q-Q Plot: Box-Muller vs CLT", xlab="Box-Muller", ylab="CLT") abline(0,1,col="red") qqplot(rv.rnorm, rv.bm, pch=".", main="Q-Q Plot: rnorm vs Box-Muller", xlab="rnorm", ylab="Box-Muller") abline(0,1,col="red") qqplot(rv.rnorm, rv.clt, pch=".", main="Q-Q Plot: rnorm vs CLT", xlab="rnorm", ylab="Box-Muller") abline(0,1,col="red")
#overlay estimated densities par(mfrow=c(1,1),lwd=2) plot(density(rv.rnorm), main="Estimated PDFs", col = "red") lines(density(rv.bm), col="green") lines(density(rv.clt), col="blue") legend(-4,.4,legend=c("rnorm", "Box-Muller", "CLT"), lty=1, lwd=2, col=c("red","green","blue"))

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