set.seed(12345)

n<-10000;

z1<-rnorm(n,0,1)
z2<-rnorm(n,0,1)
z3<-rnorm(n,0,1)
z4<-rnorm(n,0,1)
z5<-rnorm(n,0,1)

chi5<-z1^2+z2^2+z3^2+z4^2+z5^2
(mean(chi5))
(var(chi5))
hist(chi5,  freq=F, col="grey", xlab="", xlim=c(0, 25), breaks=100)
par(new=T)
plot(density(chi5), axes=F, main="", xlim=c(0, 25), lwd=2, col="blue")