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###          Extreme value theory - Exercise sheet 1	                ###
###          SS 2017			   Dr. Jürgen Kampf       	    ###
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### Exercise 3
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z=1000		# Number of repeatations of the random experiment

n=1
n=2
n=10
n=1000
n=10^5

RandomNumbers=array(rnorm(n*z),dim=c(z,n))
			# Generate an array of normal distributed random numbers.
Maxima=apply(RandomNumbers,1,max)
			# Compute the row-wise maximum.
plot(ecdf(Maxima))
plot(density(Maxima))
			# Plot empirical cumulative distribution functionand an 
			# density estimate of the sample
			# in order to get an approximation of the cumulative distribution
			# function and the density. 

# We observe that the numbers on the x-axis get larger and larger, while the shape of the 
# empirical cumulative distribution function hardly changes. So the Gaussian distribution 
# lies in the MDA of the Gumbel distributions whose shape is very similar to that of the
# Gaussian distribution.
# At the density plot we observe however an asymmetry -- we have outliers to the 
# right, but not to the left.