remove(list=ls())
set.seed(13)
ek<-function(k,x,len){
		if(k==0){sqrt(1/len)}
		else{
			if(k%%2==1){
				sqrt(2/len)*sin(pi*(k+1)*x/len)			
				}
			else{
				sqrt(2/len)*cos(pi*k*x/len)
			}
		}
}

vek<-Vectorize(ek,c("k","x"))

fNhat<-function(N,dat,x,l=(max(dat)-min(dat))){
	n<-length(dat)
	sum_p1<-outer(0:N,dat,vek,len=l)
	sum_p2<-sapply(0:N,ek,x=x,len=l)
	sum_p2<-outer(sum_p2,rep(1,n))
	sum_prod<-sum_p1*sum_p2
	sum_ges<-sum(sum_prod)
	sum_ges/n
}
dat<-runif(500)
N<-20
x<-seq(from=min(dat),to=max(dat),len=100)
res<-sapply(x,fNhat,N=N,dat=dat)
hist(dat,freq=FALSE)
lines(x,res)


####Aufgabe 2####
student<-read.table("C:/Users/Chris/Documents/Stochastik III/R/student.csv",header=T,sep=";")
n<-1:length(student[,1])
M<-5000
alpha<-0.05
corstar<-1:M
for(m in 1:M){
	index<-sample(n,30,replace=TRUE)
	studentstar<-student[index,]
	corstar[m]<-cor(studentstar$Uebung,studentstar$Klausur)
}
var(corstar)
hist(corstar,freq=FALSE)
quantile(corstar,probs=c(alpha/2,1-alpha/2))