############
# Blatt 10 #
############

#aufgabe 4
#b

p<-c(0.10, 0.25, 0.50, 0.75, 0.90)

bed_wsk<-function(p, x){
return(exp(-0.015 * x) * p / (exp(-0.015 * x) * p + exp(-0.02 * x) * (1-p)))
}

plot(0, 0, type = "n", ylim = c(0,1), xlim = c(0,1500), xlab = "Meter", ylab = "Wahrscheinlichkeit")

for(i in 1:5){
lines(seq(0,1500, 0.01), bed_wsk(p[i], seq(0,1500,0.01)), type = "l", col = i)
}
legend(800, 0.4, p, col =1:5, pch =19)


#aufgabe 5

gumb<-function(x) exp(-exp(-x))

frech<-function(x) exp(-(1+0.05 * x)^(-20))

plot(gumb, ylim = c(0.85,1), xlim = c(2,20), ylab = "Dichte")
lines(seq(2,20, 0.01), frech(seq(2,20, 0.01)), type = "l", col = "blue")

plot(seq(2,20, 0.01), gumb(seq(2,20, 0.01))-frech(seq(2,20, 0.01)), type = "l",xlab ="x",  ylab = "Differenz")

seq(2, 20, 0.01)[gumb(seq(2,20, 0.01))-frech(seq(2,20, 0.01)) == max(gumb(seq(2,20, 0.01))-frech(seq(2,20, 0.01)))]