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#    Loesungen Blatt 6     #
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# Aufgabe 1)

# 1a) Der exakte Wert ist 1 - 1/e^2

exact = 1 - 1/exp(2)
exact


# 1b)

N = 50000
x = rnorm(N)
F = sqrt(2*pi) * x * ifelse(x>0 & x<2, 1, 0)
I = sum(F) / N
I


# 1c)

N = 50000
x = runif(N, min=0, max=2)
F = 2*x*exp(-x^2/2)
I = sum(F) / N
I




# Aufgabe 2)

# 2b) 

exact = sqrt(2*pi*exp(1)) * ( 1 - pnorm(1) )
exact


# 2c) 

N = 50000
x = rnorm(N)
F = sqrt(2*pi) * exp(-x) * ifelse(x>0, 1, 0)
I = sum(F) / N
I


# 2d) 

N = 50000
x = rexp(N, rate=1)
F = exp(-x^2/2)
I = sum(F) / N
I