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#   Loesungen Blatt 7    #
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# Aufgabe 1)

# 1a) Der exakte Wert ist (1 - 1/e^4)/2

exact = (1 - 1/exp(4))/2
exact


# 1b)

N = 50000
x = rnorm(N, mean=0, sd=1/sqrt(2) )
F = sqrt(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)
I = sum(F) / N
I




# Aufgabe 2)

# 2a)

exact = sqrt(pi)*( pnorm(2,mean=0,sd=1/sqrt(2)) - pnorm(0,mean=0,sd=1/sqrt(2)) )
exact


# 2b)

N = 50000
x = rnorm(N,mean=0,sd=1/sqrt(2))

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


# 2c)

N = 50000
x = runif(N,min=0,max=2)

F = 2*exp(-x^2)
I = sum(F)/N
I




# Aufgabe 3b)

N = 50000
x = rnorm(N)

par(mfrow=c(3,3))

for(lam in 1:9)
{
  F = exp(lam*x) * exp(-lam^2/2)
  I = cumsum(F)/(1:N)
  plot(1:N, I, ylim=c(0,2), main = paste("lambda =",lam) )
  points(1:N, rep(1,N), col="red", cex = 0.5)
}


# cex = character expansion, cex = 1.0 ist normale Groesse