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# Aufgabe 1 #
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# 1a)

mycol = rainbow(10)
curve( dchisq(x,df=1), xlim=c(0,12), ylim=c(0,0.6),
       col=mycol[1], xlab="xi", ylab="densities",
       main="Dichte der chi^2-Verteilung mit n=1,2,...,10 Freiheitsgraden" )
for(i in 2:10)
{
  curve(dchisq(x,df=i),add=TRUE,col=mycol[i])
}


# 1b-d)

n = 5
N = 10000
x = rnorm(n*N)
Phi = matrix(x,nrow=N,ncol=n,byrow=TRUE)
PhiSquared = Phi*Phi
xi = rowSums(PhiSquared)
hist(xi,prob=TRUE,breaks=50)
curve(dchisq(x,df=n),add=TRUE,col="red")


# 1e)

# let's check for all n=1,2,...,10:

N=10000
for(n in 1:10)
{
  x = rnorm(n*N)
  Phi = matrix(x,nrow=N,ncol=n,byrow=TRUE)
  PhiSquared = Phi*Phi
  xi = rowSums(PhiSquared)
  info = paste("n =",n)
  hist(xi,prob=TRUE,breaks=50,xlim=c(0,30),ylim=c(0,0.5),main=info)
  curve(dchisq(x,df=n),add=TRUE,col="red")
  # wait a bit until next plot is shown:
  readline(prompt = "press Enter to continue..")
}