###  Distribution of order statistics
rm(list=ls())

n<-1000
nreps<-1000
k<-1
y<-NULL
for (i in 1:nreps) {
  x<-runif(n,0,1)
  y[i]<-x[order(x)][k]
}

hist(y,freq=FALSE)
t<-c(1:1000)/1000
lines(t,dbeta(t,k,n+1-k))




#####  Convergence in distribution

n<-100
nreps<-1000
#k<-10
y<-z<-NULL
for (i in 1:nreps) {
  x<-runif(n,0,1)
  y[i]<-max(x)  #[order(x)][k]
  z[i]<-n*(1-y[i])
}

hist(y,freq=FALSE)
t<-c(1:1000)/1000
k<-n
lines(t,dbeta(t,k,n+1-k))

hist(z,freq=FALSE)
t<-c(1:10000)/1000
lines(t,dexp(t,1))




######  Central Limit Theorem

n<-100
nreps<-1000
xbar<-NULL
for (i in 1:nreps) {
  x<-rexp(n,.1)
  xbar[i]<-mean(x)
}

mu<-10
sigma2<-100
hist(sqrt(n)*(xbar-mu)/sqrt(sigma2),freq=FALSE)
t<-c(-300:300)/100
lines(t,dnorm(t,0,1))




#######  Lab questions
n<-10000
u<-runif(n,0,1)
x<--5*(log(1-u))

z<-NULL
for (i in 1:n) {
  u<-runif(3,0,1)
  x1<--2*log(1-u)
  z[i]<-sum(x1)
}

y<-NULL
for (i in 1:n) {
  u<-runif(3,0,1)
  x1<--1.5*log(1-u)
  y[i]<-sum(x1)
}

mean(x>y)
mean(y>z)
mean(x>z)

mean(y)
mean(x)
mean(z)


hist(x,freq=FALSE)
t<-c(1:10000)/100
lines(t,exp(-t/5)/5,col=2)


n<-10000


hist(y,freq=FALSE)
t<-c(1:1000)/10
lines(t,dchisq(t,df=6))

lines(t,dgamma(t,3,1.5))



hist(x)
hist(z)